Research ArticleOn Probabilistic Alpha-Fuzzy Fixed Points andRelated Convergence Results in Probabilistic Metric andMenger Spaces under Some Pompeiu-Hausdorff-LikeProbabilistic Contractive Conditions

M De la Sen

Institute of Research and Development of Processes IIDP Faculty of Science and Technology University of the Basque CountryBarrio Sarriena Biscay 48940 Leioa Spain

Correspondence should be addressed to M De la Sen manueldelasenehues

Received 10 August 2015 Revised 24 September 2015 Accepted 13 October 2015

Academic Editor Pasquale Vetro

Copyright copy 2015 M De la SenThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the framework of complete probabilistic metric spaces and in particular in probabilistic Menger spaces this paper investigatessome relevant properties of convergence of sequences to probabilistic 120572-fuzzy fixed points under some types of probabilisticcontractive conditions

1 Introduction

Fixed point theory is an important tool to investigate the con-vergence of sequences to limits and unique limits in metricspaces and normed spaces See for instance [1ndash34] and thewide list of references cited in those papers In particularfixed point theory is also a relevant tool to investigate iterativeschemes and stability theory of continuous-time anddiscrete-time dynamic systems boundedness of the trajectory solu-tions stability of equilibrium points convergence to stableequilibrium points and the existence oscillatory solutiontrajectories [15ndash17 23 35] See also references therein Onthe other hand fixed point theory is nowadays receivingimportant research attention in the framework of probabilis-tic metric spaces See for instance [25 26 28 33 34 36ndash38] and references therein It has also to be pointed outthat Menger probabilistic metric spaces are a special classof the wide class of probabilistic metric spaces which areendowed with a triangular norm [25 26 28 33 34 36 37]and which are very useful in the context of fixed pointtheory Note that the triangular norm plays a close role tothat of the norm in normed spaces In probabilistic metricspaces the deterministic notion of distance is revisited asbeing probabilistic in the sense that given any two points 119909

and 119910 of a certain metric space a measure of the distancebetween them is a probabilistic metric 119865

119909119910(119905) rather than

the deterministic distance 119889(119909 119910) which is interpreted as theprobability of the distance between 119909 and 119910 being less than119905 (119905 gt 0) [33 34]

Fixed point theorems in complete Menger spaces forprobabilistic concepts of 119861 and 119862-contractions can be foundin [33] together with a new notion of contraction referredto as (Ψ 119862)-contraction Such a contraction was proved tobe useful for multivalued mappings while it generalizes theprevious concept of 119862-contraction On the other hand fuzzymetric spaces have been investigatedmore recently and somead hoc versions of fixed point theorems have been obtainedin that framework See for instance [4ndash6 32] and somereferences therein

This paper investigates some relevant properties of con-vergence of sequences to the so-called and defined proba-bilistic 120572-fuzzy fixed points under some types of probabilisticcontractive conditions The concept of probabilistic 120572-fuzzyfixed point is defined as an ldquoad hocrdquo conceptual extensionof that of 120572-fuzzy fixed points of [6 32] and it is oriented tothe derivation of convergence properties of fuzzy mappingsdefined on probabilistic metric spaces and in particular inprobabilistic Menger spaces

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 213174 12 pageshttpdxdoiorg1011552015213174

2 Journal of Function Spaces

Notation Preliminaries and Some Basic Concepts Denote byR+= 119911 isin R 119911 gt 0 R

0+= R

+cup 0 Z

+= 119911 isin

Z 119911 gt 0 Z0+

= Z+cup 0 119899 = 1 2 119899 and denote

by Δ F (a common used name for this class being 119863+) theset of probability distribution functions 119865 R rarr [0 1][1] which are nondecreasing and left-continuous such that119865(0) = inf

119905isinR119865(119905) = 0 and sup119905isinR119865(119905) = 1 Let 119883 be a

nonempty set and let the probabilistic metric (or distance)F 119883 times 119883 rarr Δ F be a symmetric mapping from 119883 times 119883

to Δ F where 119883 is an abstract set to the set of distancedistribution functions Δ F of the form 119865 R rarr [0 1] whichare functions of elements 119865

119909119910for every (119909 119910) isin 119883times119883 Then

the ordered pair (119883 F) is a probabilistic metric space (PM)[16 28 33 34 36ndash38] if the following constraints hold

(1) forall119909 119910 isin 119883 ((119865119909119910(119905) = 1 forall119905 isin R

+) hArr (119909 = 119910))

equivalently

119865119909119910

(119905) = 119865 (119905) lArrrArr 119909 = 119910 (1)

where 119865 isin Δ F is defined by

119865 (119905) =

0 if 119905 le 0

1 if 119905 gt 0(2)

(2) 119865119909119910(119905) = 119865

119910119909(119905) forall119909 119910 isin 119883 forall119905 isin R

(3)

forall119909 119910 119911 isin 119883 forall1199051 1199052isin R

+

((119865119909119910

(1199051) = 119865

119910119911(1199052) = 1) 997904rArr (119865

119909119911(1199051+ 119905

2) = 1))

(3)

A particular distance distribution function 119865119909119910

isin Δ F is aprobabilistic metric (or distance) which takes values 119865

119909119910(119905)

identified with a probability distance density function 119865

R rarr [0 1] in the set of all the distance distribution functionsΔ F

A Menger PM-space is a triplet (119883 F Δ) where (119883 F) isa PM-space which satisfies

119865119909119910

(1199051+ 119905

2) ge Δ (119865

119909119911(1199051) 119865

119911119910(1199052))

forall119909 119910 119911 isin 119883 forall1199051 1199052isin R

0+

(4)

under Δ [0 1] times [0 1] rarr [0 1] which is a 119905-norm (ortriangular norm) belonging to the set T of 119905-norms whichsatisfy the following properties

(1) Δ(119886 1) = 119886(2) Δ(119886 119887) = Δ(119887 119886)(3) Δ(119888 119889) ge Δ(119886 119887) if 119888 ge 119886 119889 ge 119887(4)

Δ (Δ (119886 119887) 119888) = Δ (119886 Δ (119887 119888)) (5)

A property which follows from the above ones is Δ(119886 0) = 0

for 119886 isin [0 1] Typical continuous 119905-norms are the minimum119905-norm defined by Δ

119872(119886 119887) = min(119886 119887) the product 119905-norm

defined by Δ119875(119886 119887) = 119886 sdot 119887 and the Lukasiewicz 119905-norm

defined by Δ119871(119886 119887) = max(119886 + 119887 minus 1 0) which are related

by the inequalities Δ119871le Δ

119875le Δ

119872

(i) The triplet (119883 F Δ) is a Menger space where (119883 F)is a PM-space and Δ [0 1] times [0 1] rarr [0 1] is atriangular norm which satisfies the inequality119865

119909119911(119905+

119904) ge Δ(119865119909119910(119905) 119865

119910119911(119904)) forall119909 119910 119911 isin 119883 forall119905 119904 isin R

+

(ii) Δ119872

[0 1] times [0 1] rarr [0 1] is the minimumtriangular norm defined by Δ

119872(119886 119887) = min(119886 119887)

(iii) A sequence 119909119899 sube 119883 in a probabilistic space (119883 F) is

said to be

(1) convergent to a point 119909 isin 119883 denoted by 119909119899 rarr

119909 (as) if for every 120576 isin R+and 120582 isin (0 1) there

exists some119873 = 119873(120576 120582) isin Z0+

such that

119865119909119899119909(120576) gt 1 minus 120582 forall119899 (isin Z

0+) ge 119873 (6)

(2) Cauchy if for every 120576 isin R+and 120582 isin (0 1) there

exists some119873 = 119873(120576 120582) isin Z0+

such that

119865119909119899119909119898

(120576) gt 1 minus 120582 forall119899119898 (isin Z0+) ge 119873 (7)

A PM-space (119883 F) is complete if every Cauchy sequence isconvergent

2 Concepts and Results on Probabilistic120572-Fuzzy Fixed Points

Let 119860 119861 be nonempty subsets of an abstract nonempty set119883 Then the probabilistic point-to-set distance mapping F

119883 times 119860 rarr Δ F from 119883 to 119860 denoted by 119865119909119860(119905) and the

probabilistic set-to-set distance mapping F 119860 times 119861 rarr Δ Ffrom 119860 to 119861 are respectively defined by

119865119909119860

(119905) = sup (119865119909119910

(119905) 119910 isin 119860) 119909 isin 119883 119905 isin R

119865119860119861

(119905) = sup (119865119909119910

(119905) 119909 isin 119860 119910 isin 119861) 119905 isin R(8)

The Pompeiu-Hausdorff-like probabilistic set-to-set distanceis defined by

119867119860119861

(119905) = min(inf119909isin119860

119865119909119861

(119905) inf119910isin119861

119865119910119860

(119905)) forall119905 isin R (9)

Note that 119865119909119878(0) = 0 and 119865

119878119882(0) = 0 since 119867(0) = 0 If

119860 119861 isin 119862119861(119883) where 119862119861(119883) is the set of all nonempty closedbounded subsets of 119883 then sup

119905isinR119867119860119861(119905) = sup

119909isin119860119865119909119861(119905) =

sup119910isin119861

119865119910119860(119905) = 1

A fuzzy set 119860 in 119883 is a function from 119883 to [0 1] whosegrade of membership of 119909 in 119860 is the function-value 119860(119909)The 120572-level set of 119860 is denoted by [119860]

120572defined by

[119860]120572= 119909 isin 119883 119860 (119909) ge 120572 sube 119883 if 120572 isin (0 1]

[119860]0= 119909 isin 119883 119860 (119909) gt 0 sube 119883

(10)

where 119861 denotes the closure of 119861 Let F(119883) be the collectionof all fuzzy sets in a PM-space (119883 F) Let 119879 119883 rarr F(119883) bea fuzzy mapping from an arbitrary set 119883 to F(119883) which is afuzzy subset in 119883 times 119883 and the grade of membership of 119910 in119879(119909) is 119879(119909)(119910)

Journal of Function Spaces 3

For 119860 119861 isin F(119883) 119860 sub 119861 means 119860(119909) le 119861(119909) forall119909 isin 119883Note also that if 120572 isin [120573 1] and 120573 isin (0 1] then [119860]

120572sube [119860]

120573

If there exists 120572 isin [0 1] such that [119860]120572 [119861]

120572isin 119862119861(119883) then

define

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905)

= min( inf119886isin[119879119909]

119865119886[119879119910]

120572(119910)

(119905) inf119887isin[119879119910]

119865119887[119879119909]

120572(119909)

(119905))

forall119905 isin R

(11)

The collection of all the approximate quantities in a metriclinear space 119883 is denoted by119882(119883) 119879 119883 rarr F(119884) is a fuzzymapping from an arbitrary set 119883 to F(119884) which is a fuzzysubset in119883times119884 and the grade of membership of 119910 in 119879(119909) is119879(119909)(119910)

The notation 119891 119883 | 119884 rarr 119885 means that the domain ofthe function 119891 from119883 to 119885 is restricted to the subset 119884 of119883

The next definition characterizes probabilistic fuzzy fixedpoints in an appropriate way to establish some results of thispaper

Definition 1 If F(119883) is the collection of all fuzzy sets in thePM-space (119883 F) where119883 is a nonempty abstract set and 119879

119883 rarr F(119883) is a fuzzy mapping then 119909 isin 119883 is a probabilistic120572-fuzzy fixed point of 119879 119883 rarr F(119883) if for some 120572 isin (0 1][119879119909]

120572isin 119862119861(119883) and 119909 isin [119879119909]

120572 that is (119879119909)(119909) ge 120572

Note that if 119883 is a nonempty abstract set (119883 F) is a PM-space 119860 isin F(119883) and for some 120572 isin (0 1] [119860]

120572isin 119862119861(119883)

119879 119883 rarr F(119883) then

(1) 119865[119860]120572[119860]120572

(119905) = 1 forall119905 isin R+

(2) if 119909 isin 119883 is a probabilistic 120572-fuzzy fixed point of 119879

119883 rarr F(119883) then 119865119909[119879119909]

120572

(119905) = 1 forall119905 isin R+

(3) if [119879119909]120572(119909)

isin 119862119861(119883) and 119909 isin 119883 is not a probabilistic120572-fuzzy fixed point of 119879 119883 rarr F(119883) then 119909 notin

[119879119909]120572(119909)

equivalently (119879119909)(119909) lt 120572 and 119865119909[119879119909]

120572

(119905) lt

1 119905 isin [0 1199051] for some 119905

1= 119905

1(119909) isin R

+

The following result holds

Theorem 2 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions hold

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(12)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 (119909 119910 119905) =

3

sum

119894=1

119886119894(119909 119910 119905) le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(13)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(14)

Then a sequence 119909119899may be built for any given arbitrary 119909

1=

119909 isin 119883 satisfying 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) with 120572(119909

119899) sube

(0 1] satisfying lim119899rarrinfin

119865119909119899119909119899+1

(119905) = 1 forall119905 isin R+

If in addition (119883 F) is endowed with the minimumtriangular norm Δ

119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Take arbitrary points 1199091= 119909 isin 119883 119909

2= 119910 isin [119879119909

1]120572(1199091)

for some given existing 120572(1199091) isin (0 1] such that [119879119909

1]120572(1199091)is

nonempty and take also some existing120572(1199092) isin (0 1] such that

[1198791199092]120572(1199092)is nonempty Note that since 119865

119909119860(119905) = sup(119865

119909119910(119905)

119910 isin 119860) for any 119909 isin 119883 and 119905 isin R then 1198651199091[1198791199091]120572(1199091)

(119905) ge

11986511990911199092

(119905) Thus one gets from the contractive condition (12)that

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905)

= min( inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905)

inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)) ge 1198861(119909

1 119909

2 119896119905)

sdot 1198651199091[1198791199091]120572(119909)

(119905) + 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119905)

+ 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905) ge 1198861(119909

1 119909

2 119896119905)

sdot 1198651199091[1198791199091]120572(1199091)

(119905) + 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

+ 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905) = (1198861(119909

1 119909

2 119896119905)

+ 1198863(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905) + 1198862(119909

1 119909

2 119896119905)

sdot 1198651199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

(15)

for any given 120572(1199092) isin (0 1] since 119865

1199091[1198791199091]120572(1199091)

(119905) ge 11986511990911199092

(119905)

for all 119905 isin R+since 119909

2isin [119879119909

1]120572(1199091) Then again since 119909

2isin

[1198791199091]120572(1199091) the following cases can arise for each 119905 isin R

+

4 Journal of Function Spaces

Case (a) One has119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905)

= min( inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905) 1198651199092[1198791199092]120572(1199092)

(119896119905))

le 1198651199092[1198791199092]120572(1199092)

(119896119905)

(16)

for some given 119905 isin R+ Thus from (15) and (16) one gets

1198651199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

(17)

and one gets for the given 119905 isin R+that since 119867(119905) is

nondecreasing and left-continuous then

119865119909[119879119909]

120572(119909)

(119905) ge 119865119909[119879119909]

120572(119909)

(119896119905) forall119905 isin R+ forall119909 isin 119883 (18)

and since 119896 isin (0 1) one gets from (17) that

1198651199092[1198791199092]120572(1199092)

(119905) ge 1198651199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

(19)

and then since [1198791199092]120572(1199092)is closed and nonempty there

exists 1199093isin [119879119909

2]120572(1199092)such that from (19) and the fact that

1198651199092[1198791199092]120572(1199092)

(119905) ge 1198651199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

11986511990921199093

(119905) ge 11986511990921199093

(119896119905) = 1198651199092[1198791199092]120572(1199092)

(119896119905)

ge

1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)

1 minus 1198862(119909

1 119909

2 119896119905)

11986511990911199092

(119905)

forall119905 isin R+

(20)

and equivalently

11986511990921199093

(119905) ge 119892 (1199091 119909

2 119905) 119865

11990911199092

(119896minus1

119905) forall119905 isin R+ (21)

where 119892(1199091 119909

2 119905) = (119886

1(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905))(1 minus 119886

2(119909

1

1199092 119905)) forall119905 isin R

+

Case (b) One has119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)

= min( inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905) 1198651199091[1198791199091]120572(1199091)

(119896119905))

le 1198651199091[1198791199091]120572(1199091)

(119896119905)

(22)

and some 119905 isin R+and 119909

3isin [119879119909

2]120572(1199092)can be chosen for

the previously taken 1199092isin [119879119909

1]120572(1199091)so that 119865

11990921199093

(119896119905) =

inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905) Thus

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)

= 11986511990921199093

(119896119905)

(23)

and since 1198651199092[1198791199092]120572(1199092)

(119896119905) ge 11986511990921199093

(119896119905) one gets for the given119905 isin R

+

11986511990921199093

(119905) ge 11986511990921199093

(119896119905)

ge 1198861(119909

1 119909

2 119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198863(119909

1 119909

2 119905) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119905) 119865

11990921199093

(119896119905)

(24)

Then one gets from (24) that

(1 minus 1198862(119909

1 119909

2 119905)) 119865

11990921199093

(119896119905)

ge (1198861(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905)) 119865

11990911199092

(119905)

(25)

which implies (21) So from Cases (a)-(b) for each given119905 isin R

+ and 119909

1isin 119883 there exist 120572(119909

1) 120572(119909

2) isin (0 1] and

points 1199092isin [119879119909

1]120572(1199091)and 119909

2isin [119879119909

2]120572(1199092)in nonempty level

sets [1198791199091]120572(1199091)and [119879119909

2]120572(1199092)such that (22) holds Proceeding

recursively one gets that a sequence 119909119899may be built for any

arbitrary 1199091= 119909 isin 119883 and 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] forall119899 isin Z

+which satisfies the recursion

11986511990921199093

(119905) ge 119892 (1199091 119909

2 119905) 119865

11990911199092

(119896minus1

119905) forall119905 isin R+

11986511990931199094

(119905) ge 119892 (1199092 119909

3 119905) 119865

11990921199093

(119896minus1

119905)

ge 119892 (1199092 119909

3 119905) 119892 (119909

1 119909

2 119896

minus1

119905) 11986511990911199092

(119896minus2

119905)

forall119905 isin R+

119865119909119899+1119909119899+2

(119905) ge [

119899

prod

119894=1

119892 (119909119894 119909

119894+1 119896

minus119894+1

119905)] 11986511990911199092

(119896minus119899

119905)

forall119905 isin R+ forall119899 isin Z

+

(26)

where

0 lt 119892 (119909119894 119909

119894+1 119905) =

1198861(119909

119894 119909

119894+1 119905) + 119886

3(119909

119894 119909

119894+1 119905)

1 minus 1198862(119909

119894 119909

119894+1 119905)

le 1 forall119905 isin R+ forall119894 isin Z

+

(27)

Note that since0 lt 119886 (119909

119894 119909

119894+1 119905) le 1

0 le 1198862(119909

119894 119909

119894+1 119905) lt 1

forall119905 isin R+ forall119894 isin Z

+

(28)

119899

prod

119894=0

[

1198861(119909

119894 119909

119894+1 119896

minus119894+1

119905) + 1198863(119909

119894 119909

119894+1 119896

minus119894+1

119905)

1 minus 1198862(119909

119894 119909

119894+1 119896

minus119894+1119905)

]

997888rarr 1 as 119899 997888rarr infin forall119905 isin R+

(29)

Journal of Function Spaces 5

then 120575(Γ1119905)120575(Γ

0119905) = +infin forall119905 isin R

+ forall119894 isin Z

+ where 120575(Γ

0119905) =

prodinfin

119895=0[1205750

119895119905] = prod

infin

119895=0[1 minus 120575

1

119895119905] and 120575(Γ

1119905) = prod

infin

119895=0[1205751

119895119905] = prod

infin

119895=0[1 minus

1205750

119895119905] forall119905 isin R

+are discrete measures of the subsequent sets

Γ0119905= Γ

0119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) isin [0 1)

forall119905 isin R+

Γ1119905= Γ

1119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) = 1

forall119905 isin R+

(30)

where 1205750119895119905and 1205751

119895119905are Dirac measures defined by

1205750

119895119905= 1 minus 120575

1

119895119905=

1 if 119895 isin Γ0119905

0 if 119895 notin Γ0119905

forall119905 isin R+

1205751

119895119905= 1 minus 120575

0

119895119905=

1 if 119895 isin Γ1119905

0 if 119895 notin Γ1119905

forall119905 isin R+

(31)

Then one gets from (26) (29) and lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) =

lim120591rarr+infin

minus119867(120591) = 119867(+infinminus

) = 1 forall119905 isin R+ since 119896 isin (0 1)

that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+with 119909

119899+1isin [119879119909

119899]120572(119909119899)

forall119899 isin Z+since

lim119899rarrinfin

119865119909119899+1119909119899+2

(119905) ge ( lim119899rarrinfin

[

119899

prod

119894=1

119892 (119909119894 119909

119894+1 119896

minus119894+1

119905)])

sdot ( lim119899rarrinfin

11986511990911199092

(119896minus119899

119905)) forall119905 isin R+

(32)

Since lim119899rarrinfin

119865119909119899+1119909119899

(119905) = lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) = 1 forall119905 isin

R+and any given 119909

1isin 119883 then for any given 120576 isin R

+

and 120582 isin (0 1) there is 119873 = 119873(120576 120582) isin Z0+

such that119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+ Assume on the contrary

that there exist 119899119896

ge 119873 119899119896+2

gt 119899119896+1

gt 119899119896such that

119865119909119899119896+119894+1

119909119899119896+119894

(120576) ge 119865119909119899119896+119894+1

119909119899119896+119894

(1205762) gt 1 minus 120582 for 119894 = 0 1 and1minus120582 ge 119865

119909119899119896+2

119909119899119896

(120576)Then one has the following contradictionfor the subsequence 119909

119899119896

of 119909119899

1 minus 120582 ge 119865119909119899119896+2

119909119899119896

(120576)

ge Δ119872(119865

119909119899119896+2

119909119899119896+1

(

120576

2

) 119865119909119899119896+1

119909119899119896

(

120576

2

))

gt 1 minus 120582

(33)

Then 119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+

so that 119909119899 is

a Cauchy sequence Since (119883 F Δ119872) is complete one gets

119909119899 rarr 119909

lowast and 120572(119909119899) rarr 120572(119909

lowast

)

It is now proved that 119909lowast isin [119879119909lowast

]120572(119909lowast) Assume on the

contrary that 119909lowast notin [119879119909lowast

]120572(119909lowast) that is (119879119909lowast)(119909lowast) lt 120572(119909

lowast

)

and for some given 119910 isin [119879119909lowast

]120572(119909lowast) there is 119905

1= 119905

1(119910) isin R

+

such that 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) lt 1 for 119905 isin [0 1199051] Then since

119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+and since 119910 = 119909

lowast

1 gt 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119909lowast119910(119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

) 119865119909119899+1119910(

119905

4

)))

= Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

119905

4

) 119865[119879119909119899]120572(119909119899)

119910(

119905

4

)))

119905 isin [0 1199051]

(34)

If 119910 isin [119879119909lowast

]120572(119909lowast)is chosen to fulfill 119865

[119879119909119899]120572(119909119899)

119910(11990514) =

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(11990514) then

1 gt lim sup119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

1199051

2

) Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

119865[119879119909119899]120572(119909119899)

119910(

1199051

4

))) ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

1199051

2

)

Δ119872( lim119899rarrinfin

119865119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

))) = Δ119872(1

Δ119872(1 lim sup

119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)))

= lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)

= lim119899rarrinfin

119865[119879119909lowast]120572(119909lowast)[119879119909lowast]120572(119909lowast)

(

1199051

4

) = 1

for some 1199051isin R

+

(35)

a contradiction Then 119909lowast isin [119879119909lowast]120572(119909lowast)with 120572lowast = 120572(119909

lowast

) thatis it is a probabilistic 120572lowast-fuzzy fixed point of 119879 119883 rarr F(119883)

Corollary 3 Let F(119883) be the collection of all fuzzy sets ina PM-space (119883 F) where 119883 is a nonempty abstract set andlet 119879 119883 rarr F(119883) be a fuzzy mapping Assume that thecontractive condition (12) holds for some real constant 119896 isin

(0 1) subject to condition (2) of Theorem 2 and the specificparticular form of condition (3)

6 Journal of Function Spaces

(31015840

) there exists and strictly increasing sequence of nonnegativeintegers 119873

119899 which satisfies

119873119899+2minus1

prod

119895=119873119899+1

[

1198862(119909 119910 119896

minus119894+1

119905) + 1198863(119909 119910 119896

minus119894+1

119905)

1 minus 1198862(119909 119910 119896

minus119894+1119905)

] =

1

prod119873119899+1minus1

119895=119873119899

[(1198862(119909 119910 119896

minus119894+1119905) + 119886

3(119909 119910 119896

minus119894+1119905)) (1 minus 119886

2(119909 119910 119896

minus119894+1119905))]

forall119905 isin R+ forall119909 isin 119883 119910 isin [119879119909]

120572(119909)sub 119862119861 (119883) forall119899 isin Z

0+

(36)

Proof It is direct from that of Theorem 2 since condition(31015840) guarantees condition (3) of Theorem 2 for any finite firstelement 119873

0isin Z

0+of a strictly increasing sequence 119873

119899

subject to 0 lt 1198721le 119873

119899+1minus 119873

119899le 119872

2lt +infin

Corollary 4 Theorem 2 and Corollary 3 also hold if thecontractive condition (1) is modified as follows

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905)

ge 1198861(119909

1 119909

2 119896119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) + 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905)

= (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

forall119905 isin R+ forall119909

1= 119909 isin 119883 119909

119899+1isin [119879119909

119899]120572(119909119899)sub 119862119861 (119883) forall119899 isin Z

0+

(37)

Proof It is direct from that of Theorem 2 since the abovecontraction condition follows from (12) as a lower-boundwhich has been used in the proof of Theorem 2

Corollary 5 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of119883 and

each sequence 119909119899 sub 119883 of the form 119909

1isin 119883 119909

119899+1isin

[119879119909119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+which satisfies for some

real constant 119896 isin (0 1) the contractive constraint

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge min (1 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905) + 1198863(119909 119910 119896119905) 119865

119909119910(119905))

(38)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894 119883 times 119883 times R

0+rarr [0 1] for

119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Then a sequence 119909119899 may be built for any given arbitrary

1199091= 119909 isin 119883 satisfying 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] and lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isin R+

If in addition (119883 F) is endowed with the minimumtriangular norm Δ

119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each above sequence 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Define the indicator function sequence 120590119894 119883 times 119883 times

R0+

rarr 0 1 119894 isin Z+as

120590119894(119909

119894 119909

119894+1 119896

minus119894+1

119905)

=

1 if 119892 (119909119894 119909

119894+1 119896

minus119894+1

119905) 119865119909119894119909119894+1

(119896minus119894

119905) le 1

0 otherwise

forall119905 isin R+

(39)

with 119892(119909119894 119909

119894+1 119905) defined as in Theorem 2 Then even if for

some 119895 isin Z+and all 119894(ge 119895) isin Z

+ 120590

119894(119909

119894 119909

119894+1 119905) = 0 because

119892(119909119894 119909

119894+1 119896

minus119894+1

119905)119865119909119894119909119894+1

(119896minus119894

119905) gt 1 forall119894(ge 119895) isin Z+ it follows

from (38)-(39) and (26) that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

with 119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+ The rest of the proof is close

to that of Theorem 2

Example 6 It is claimed through this example to revisitthe idea of fuzzy fixed point addressed in [4ndash6 32] tothat of probabilistic fuzzy fixed point according to thedefinition and the formalism given above Consider 119883 =

06 07 08 06 07 08 and let 119879 119883 rarr F(119883) be aprobabilistic fuzzy mapping defined as follows

119879 (06) (119905) = 119879 (08) (119905) =

3

4

if 119905 = 08

1

2

if 119905 = 07

0 if 119905 = 06

119879 (07) (119905) =

0 if 119905 = 08

1

3

if 119905 = 07

3

4

if 119905 = 06

(40)

The 120572-level sets are

[11987906]12

= [11987908]12

= 07 08

[11987906]34

= [11987908]34

= 08

[11987907]34

= 06

[11987907]13

= 07

(41)

Journal of Function Spaces 7

Note that 119865119909[119879119909]

120572(119909)

(119905) = sup(119865119909119910(119905) 119910 isin [119879119909]

120572(119909)) for any

119909 isin 119883 and 119905 isin R and some 120572(119909) isin (0 1] The probabilitydensity functions are as follows with 119865

119909119910(119905) = 119905(119905 + |119909 minus 119910|)

forall119909 119910 isin 119883 forall119905 isin R+

11986506[11987906]

34

(119905) = 1198650608

(119905) =

119905

119905 + 02

forall119905 isin R+

11986506[11987906]

12

(119905) = 1198650607

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

12

(119905) = 1198650807

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

34

(119905) = 1198650808

(119905) = 1 forall119905 isin R+

11986507[11987907]

34

(119905) = 1198650706

(119905) =

119905

119905 + 01

forall119905 isin R+

11986507[11987907]

13

(119905) = 1198650707

(119905) = 1 forall119905 isin R+

119867[119879119909]12[119879119909]12

(119905) =

119905

119905 + 01

for 119909 = 06 08 forall119905 isin R+

119867[119879119909]34[119879119909]34

(119905) = 1 for 119909 = 06 08 forall119905 isin R+

119867[119879119909]12[119879119910]34

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

13

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

34

(119905) =

119905

119905 + 02

for 119909 119910 = 06 08 forall119905 isin R+

119867[11987907]

13[11987907]

13

(119905) = 1 forall119905 isin R+

119867[11987907]

34[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

119867[11987907]

13[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

(42)

Assume that the contractive condition of Theorem 2 holdsunder the form

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905) ge 120572 (119909119899 119909

119899+1 119905 119899)

sdot [119865119909119899[119879119909119899]120572(119909119899)

(119896minus1

119905) + 119865119909119899+1[119879119909119899+1]120572(119909119899+1)

(119896minus1

119905)

+ 119865119909119899119909119899+1

(119896minus1

119905)] 119899 isin Z0+

(43)

for sequences 119909119899 sub 119883 with initial points 119909

0= 119909 isin 119883 119909

1isin

[1198791199090]120572 where 120572(119909 119910 119905 119899) = 120572

119894(119909 119910 119905 119899) 119894 = 1 2 3 forall119905 isin R

+

satisfies

120572 (119909 119910 119905 119899) = 120572119886(119905) le

119896119905

3 (119896119905 + 02)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

120572 (119909 119910 119905 119899) = 120572119887(119905) le

1

3

(1 minus 119890minus120582119905

)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

(44)

for some ℓ(ge 2) isin Z0+ some 119896 isin (0 1) and some 120582 isin R

+

Note that [11987907]13

= 07 with 11986507[11987907]

13

(119905) = 1198650707

(119905) =

1 forall119905 isin R+is a probabilistic 13-fuzzy fixed point of 119879

119883 rarr F(119883) to which the sequences 06 08 07 07 08 06 07 07 converge

3 Further Results

Note that the uniqueness of probabilistic fuzzy fixed points isnot an interesting property to study in the context of prob-abilistic fuzzy fixed point since distinct level sets associatedwith mappings of the form 119879 119883 rarr F(119883) can easily haveintersections of cardinal greater than one in many problemsin the fuzzy context The subsequent result gives conditionsfor the case when several distinct probabilistic fuzzy fixedpoints if they exist are in the intersections of their respectivelevel sets under slightly extended contractive conditions ofthose given in Section 2The extended conditions are of directapplicability

Theorem 7 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 with an

extended contractive condition (12) such that it also holds forany 119909 119910 isin 119883 being probabilistic 120572(119909lowast)-fuzzy fixed points119909 = 119909

lowast

isin [119879119909lowast

]120572(119909lowast)and 119910 = 119910

lowast

isin [119879119910lowast

]120572(119910lowast)of 119879 119883 rarr

F(119883) Then the set [119879119910lowast]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)is nonempty and

119909lowast

119910lowast

isin ([119879119910lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast))

Proof If119909lowast = 119910lowast the result is obvious Assume that there exista probabilistic 120572(119909lowast)-fuzzy fixed point 119909lowast isin [119879119909lowast]

120572(119909lowast)and a

probabilistic 120572(119910lowast)-fuzzy fixed point 119910lowast( = 119909lowast

) isin [119879119910lowast

]120572(119910lowast)

Consider two convergent sequences 119909119899 rarr 119909

lowast and 119910119899 rarr

119910lowast in119883 Then

119865119909119899119910119899

(119905) ge Δ119872(119865

119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

119865119910119899119910lowast (

119905

4

))) forall119905 isin R+

lim inf119899rarrinfin

119865119909119899119910119899

(119905) ge lim inf119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909lowast119910lowast (

119905

4

) 119865119910119899119910lowast (

119905

4

)))

ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

lim119899rarrinfin

119865119910119899119910lowast (

119905

4

))) = Δ119872(1 Δ

119872(119865

119909lowast119910lowast (

119905

4

) 1))

ge 119865119909lowast119910lowast (

119905

4

) forall119905 isin R+

(45)

Assume that 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+

Then from the extended contractive condition (12) and since1 = 119865

119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905) ge 119865119909lowast119910lowast(119905) forall119905 isin R

+

one gets for some 119911119909isin [119879119909

lowast

]120572(119909lowast)and 119911

119910isin [119879119910

lowast

]120572(119910lowast)

8 Journal of Function Spaces

since [119879119909lowast]120572(119909lowast)and [119879119910lowast]

120572(119910lowast)are members of 119862119861(119883) that

is nonempty closed and bounded sets

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905)

= min (119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905))

= min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge 1198861(119909

lowast

119910lowast

119896119905)

sdot 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) + 1198862(119909

lowast

119910lowast

119896119905) 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905)

+ 1198863(119909

lowast

119910lowast

119896119905) 119865119909lowast119910lowast (119905) ge (119886

1(119909

lowast

119910lowast

119896119905)

+ 1198862(119909

lowast

119910lowast

119896119905) + 1198863(119909

lowast

119910lowast

119896119905)) 119865119909lowast119910lowast (119905)

(46)

so that

min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) 119865119909lowast119910lowast (119896

minus1

119905)

ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) Δ119872(119865

119909lowast119911119910

(

119896minus1

119905

2

) 119865119911119910119910lowast (

119896minus1

119905

2

))

ge

119898

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)] Δ119872(119865

119909lowast119911119910

((2119896)minus119898

119905) 119865119911119910119910lowast ((2119896)

119898

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot lim inf119899rarrinfin

Δ119872(119865

119909lowast119911119910

((2119896)minus119899

119905) 119865119911119910119910lowast ((2119896)

minus119899

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot Δ119872(119865

119909lowast119911119910

( lim119899rarrinfin

(2119896)minus119899

119905) 119865119911119910119910lowast ( lim

119899rarrinfin

(2119896)minus119899

119905))

= lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])119867 (+infinminus

) = 1 forall119905 isin R+ forall119898 isin Z

+

(47)

and then 119910lowast = 119911119909 119909lowast = 119911

119910([119879119910

lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)) so that

[119879119910lowast

]120572(119910lowast)cap[119879119909

lowast

]120572(119909lowast)is nonemptyWe can now assume that

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) le 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+ It is direct to

prove in a close way to the above proof that 119910lowast isin [119879119910lowast]120572(119910lowast)cap

[119879119909lowast

]120572(119909lowast)

Remark 8 Note that a direct consequence of Theorem 7from the definition of the level sets is that

119909lowast

119910lowast

isin (( ⋂

120573isin(0120572(119909lowast)]

[119879119909lowast

]120573) cap ( ⋂

120573isin(0120572(119910lowast)]

[119879119910lowast

]120573))

(48)

if 119909lowast isin [119879119909lowast

]120572(119909lowast)and 119910lowast isin [119879119910

lowast

]120572(119910lowast)are any probabilistic

120572(119909lowast

) and 120572(119910lowast)-fuzzy fixed points

Remark 9 Note that corollaries to Theorem 7 can also bestated ldquomutatis-mutandisrdquo under extendedcontractiveconditions

for probabilistic fuzzy fixed points to those given in Corollar-ies 3ndash5

Theorem 10 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 and condi-

tions (1)ndash(3) where the contractive condition

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(49)

is extended to hold for any 119909 119910 isin 119883 with 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Proof Note from Theorem 2 a sequence 119909119899 sub 119883 can be

built being a (convergent) Cauchy sequence such that 119909119899 rarr

Journal of Function Spaces 9

119909lowast 119909

119899+1= 119879119909

119899isin [119879119909

119899]120572(119909119899) forall119899 isin Z

0+for any given 119909

0isin 119883

Then

lim119899rarrinfin

119865119909119899119879119909119899

(119905) = lim119899rarrinfin

119865119909119899119909lowast (119905) = 1 forall119905 isin R

+

lim119899rarrinfin

119865119879119909119899119909lowast (119905) = 1 forall119905 isin R

+

since 119865119879119909119899119909lowast (119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

)) 119865119879119909119899119879119909119899

(

119905

4

))

gt min (1 minus 120582 1 minus 120582 1) = 1 minus 120582

(50)

for any given 119905 isin R+ 120582 isin (0 1) 119899(ge 119873

0) isin Z

0+ and 119873

0=

1198730(120576 120582) ge max(119873

1 119873

2) such that

119865119909lowast119909119899

(

119905

2

) gt 1 minus 120582

for 119899 (ge 1198731) isin Z

0+and some 119873

1= 119873

1(120576 120582) isin Z

0+

119865119909119899+1119909119899

(

119905

4

) gt 1 minus 120582

for 119899 (ge 1198732) isin Z

0+and some 119873

2= 119873

2(120576 120582) isin Z

0+

(51)

since 119865119879119909119899119879119909119899

(1199054) = 1 forall119905 isin R+ forall119899 isin Z

0+from property (1)

of (3) for PM-spaces

Theorem 11 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

(52)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894isin [0 1] for 119894 =

1 3 4 5 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 =

5

sum

119894=1

119886119894le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(53)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(54)

Then a sequence 119909119899 may be constructed for any given

arbitrary 1199091= 119909 isin 119883 such that 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883)

with 120572(119909119899) sube (0 1] satisfying lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isinR+If in addition (119883 F) is endowed with the minimum

triangular norm Δ119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Since [119879119909]120572(119909)

[119879119910]120572(119910)

sub 119883

max (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min119911isin[119879119909]

120572(119909)120596isin[119879119910]

120572(119910)

(119865119911[119879119910]

120572(119910)

(119896119905) 119865120596[119879119909]

120572(119909)

(119896119905))

= 119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905) + 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

forall119905 isin R+ forall119909 119910 isin 119883

(55)

Now for any given 119909 119910 isin 119883 Then the following cases canoccur

(a) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905) forall119905 isin R+

(1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4) 119865

119910[119879119910]120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4+ 119886

5) 119865

119909119910(119905) forall119905 isin R

+

(56)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(57)

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

Notation Preliminaries and Some Basic Concepts Denote byR+= 119911 isin R 119911 gt 0 R

0+= R

+cup 0 Z

+= 119911 isin

Z 119911 gt 0 Z0+

= Z+cup 0 119899 = 1 2 119899 and denote

by Δ F (a common used name for this class being 119863+) theset of probability distribution functions 119865 R rarr [0 1][1] which are nondecreasing and left-continuous such that119865(0) = inf

119905isinR119865(119905) = 0 and sup119905isinR119865(119905) = 1 Let 119883 be a

nonempty set and let the probabilistic metric (or distance)F 119883 times 119883 rarr Δ F be a symmetric mapping from 119883 times 119883

to Δ F where 119883 is an abstract set to the set of distancedistribution functions Δ F of the form 119865 R rarr [0 1] whichare functions of elements 119865

119909119910for every (119909 119910) isin 119883times119883 Then

the ordered pair (119883 F) is a probabilistic metric space (PM)[16 28 33 34 36ndash38] if the following constraints hold

(1) forall119909 119910 isin 119883 ((119865119909119910(119905) = 1 forall119905 isin R

+) hArr (119909 = 119910))

equivalently

119865119909119910

(119905) = 119865 (119905) lArrrArr 119909 = 119910 (1)

where 119865 isin Δ F is defined by

119865 (119905) =

0 if 119905 le 0

1 if 119905 gt 0(2)

(2) 119865119909119910(119905) = 119865

119910119909(119905) forall119909 119910 isin 119883 forall119905 isin R

(3)

forall119909 119910 119911 isin 119883 forall1199051 1199052isin R

+

((119865119909119910

(1199051) = 119865

119910119911(1199052) = 1) 997904rArr (119865

119909119911(1199051+ 119905

2) = 1))

(3)

A particular distance distribution function 119865119909119910

isin Δ F is aprobabilistic metric (or distance) which takes values 119865

119909119910(119905)

identified with a probability distance density function 119865

R rarr [0 1] in the set of all the distance distribution functionsΔ F

A Menger PM-space is a triplet (119883 F Δ) where (119883 F) isa PM-space which satisfies

119865119909119910

(1199051+ 119905

2) ge Δ (119865

119909119911(1199051) 119865

119911119910(1199052))

forall119909 119910 119911 isin 119883 forall1199051 1199052isin R

0+

(4)

under Δ [0 1] times [0 1] rarr [0 1] which is a 119905-norm (ortriangular norm) belonging to the set T of 119905-norms whichsatisfy the following properties

(1) Δ(119886 1) = 119886(2) Δ(119886 119887) = Δ(119887 119886)(3) Δ(119888 119889) ge Δ(119886 119887) if 119888 ge 119886 119889 ge 119887(4)

Δ (Δ (119886 119887) 119888) = Δ (119886 Δ (119887 119888)) (5)

A property which follows from the above ones is Δ(119886 0) = 0

for 119886 isin [0 1] Typical continuous 119905-norms are the minimum119905-norm defined by Δ

119872(119886 119887) = min(119886 119887) the product 119905-norm

defined by Δ119875(119886 119887) = 119886 sdot 119887 and the Lukasiewicz 119905-norm

defined by Δ119871(119886 119887) = max(119886 + 119887 minus 1 0) which are related

by the inequalities Δ119871le Δ

119875le Δ

119872

(i) The triplet (119883 F Δ) is a Menger space where (119883 F)is a PM-space and Δ [0 1] times [0 1] rarr [0 1] is atriangular norm which satisfies the inequality119865

119909119911(119905+

119904) ge Δ(119865119909119910(119905) 119865

119910119911(119904)) forall119909 119910 119911 isin 119883 forall119905 119904 isin R

+

(ii) Δ119872

[0 1] times [0 1] rarr [0 1] is the minimumtriangular norm defined by Δ

119872(119886 119887) = min(119886 119887)

(iii) A sequence 119909119899 sube 119883 in a probabilistic space (119883 F) is

said to be

(1) convergent to a point 119909 isin 119883 denoted by 119909119899 rarr

119909 (as) if for every 120576 isin R+and 120582 isin (0 1) there

exists some119873 = 119873(120576 120582) isin Z0+

such that

119865119909119899119909(120576) gt 1 minus 120582 forall119899 (isin Z

0+) ge 119873 (6)

(2) Cauchy if for every 120576 isin R+and 120582 isin (0 1) there

exists some119873 = 119873(120576 120582) isin Z0+

such that

119865119909119899119909119898

(120576) gt 1 minus 120582 forall119899119898 (isin Z0+) ge 119873 (7)

A PM-space (119883 F) is complete if every Cauchy sequence isconvergent

2 Concepts and Results on Probabilistic120572-Fuzzy Fixed Points

Let 119860 119861 be nonempty subsets of an abstract nonempty set119883 Then the probabilistic point-to-set distance mapping F

119883 times 119860 rarr Δ F from 119883 to 119860 denoted by 119865119909119860(119905) and the

probabilistic set-to-set distance mapping F 119860 times 119861 rarr Δ Ffrom 119860 to 119861 are respectively defined by

119865119909119860

(119905) = sup (119865119909119910

(119905) 119910 isin 119860) 119909 isin 119883 119905 isin R

119865119860119861

(119905) = sup (119865119909119910

(119905) 119909 isin 119860 119910 isin 119861) 119905 isin R(8)

The Pompeiu-Hausdorff-like probabilistic set-to-set distanceis defined by

119867119860119861

(119905) = min(inf119909isin119860

119865119909119861

(119905) inf119910isin119861

119865119910119860

(119905)) forall119905 isin R (9)

Note that 119865119909119878(0) = 0 and 119865

119878119882(0) = 0 since 119867(0) = 0 If

119860 119861 isin 119862119861(119883) where 119862119861(119883) is the set of all nonempty closedbounded subsets of 119883 then sup

119905isinR119867119860119861(119905) = sup

119909isin119860119865119909119861(119905) =

sup119910isin119861

119865119910119860(119905) = 1

A fuzzy set 119860 in 119883 is a function from 119883 to [0 1] whosegrade of membership of 119909 in 119860 is the function-value 119860(119909)The 120572-level set of 119860 is denoted by [119860]

120572defined by

[119860]120572= 119909 isin 119883 119860 (119909) ge 120572 sube 119883 if 120572 isin (0 1]

[119860]0= 119909 isin 119883 119860 (119909) gt 0 sube 119883

(10)

where 119861 denotes the closure of 119861 Let F(119883) be the collectionof all fuzzy sets in a PM-space (119883 F) Let 119879 119883 rarr F(119883) bea fuzzy mapping from an arbitrary set 119883 to F(119883) which is afuzzy subset in 119883 times 119883 and the grade of membership of 119910 in119879(119909) is 119879(119909)(119910)

Journal of Function Spaces 3

For 119860 119861 isin F(119883) 119860 sub 119861 means 119860(119909) le 119861(119909) forall119909 isin 119883Note also that if 120572 isin [120573 1] and 120573 isin (0 1] then [119860]

120572sube [119860]

120573

If there exists 120572 isin [0 1] such that [119860]120572 [119861]

120572isin 119862119861(119883) then

define

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905)

= min( inf119886isin[119879119909]

119865119886[119879119910]

120572(119910)

(119905) inf119887isin[119879119910]

119865119887[119879119909]

120572(119909)

(119905))

forall119905 isin R

(11)

The collection of all the approximate quantities in a metriclinear space 119883 is denoted by119882(119883) 119879 119883 rarr F(119884) is a fuzzymapping from an arbitrary set 119883 to F(119884) which is a fuzzysubset in119883times119884 and the grade of membership of 119910 in 119879(119909) is119879(119909)(119910)

The notation 119891 119883 | 119884 rarr 119885 means that the domain ofthe function 119891 from119883 to 119885 is restricted to the subset 119884 of119883

The next definition characterizes probabilistic fuzzy fixedpoints in an appropriate way to establish some results of thispaper

Definition 1 If F(119883) is the collection of all fuzzy sets in thePM-space (119883 F) where119883 is a nonempty abstract set and 119879

119883 rarr F(119883) is a fuzzy mapping then 119909 isin 119883 is a probabilistic120572-fuzzy fixed point of 119879 119883 rarr F(119883) if for some 120572 isin (0 1][119879119909]

120572isin 119862119861(119883) and 119909 isin [119879119909]

120572 that is (119879119909)(119909) ge 120572

Note that if 119883 is a nonempty abstract set (119883 F) is a PM-space 119860 isin F(119883) and for some 120572 isin (0 1] [119860]

120572isin 119862119861(119883)

119879 119883 rarr F(119883) then

(1) 119865[119860]120572[119860]120572

(119905) = 1 forall119905 isin R+

(2) if 119909 isin 119883 is a probabilistic 120572-fuzzy fixed point of 119879

119883 rarr F(119883) then 119865119909[119879119909]

120572

(119905) = 1 forall119905 isin R+

(3) if [119879119909]120572(119909)

isin 119862119861(119883) and 119909 isin 119883 is not a probabilistic120572-fuzzy fixed point of 119879 119883 rarr F(119883) then 119909 notin

[119879119909]120572(119909)

equivalently (119879119909)(119909) lt 120572 and 119865119909[119879119909]

120572

(119905) lt

1 119905 isin [0 1199051] for some 119905

1= 119905

1(119909) isin R

+

The following result holds

Theorem 2 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions hold

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(12)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 (119909 119910 119905) =

3

sum

119894=1

119886119894(119909 119910 119905) le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(13)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(14)

Then a sequence 119909119899may be built for any given arbitrary 119909

1=

119909 isin 119883 satisfying 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) with 120572(119909

119899) sube

(0 1] satisfying lim119899rarrinfin

119865119909119899119909119899+1

(119905) = 1 forall119905 isin R+

If in addition (119883 F) is endowed with the minimumtriangular norm Δ

119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Take arbitrary points 1199091= 119909 isin 119883 119909

2= 119910 isin [119879119909

1]120572(1199091)

for some given existing 120572(1199091) isin (0 1] such that [119879119909

1]120572(1199091)is

nonempty and take also some existing120572(1199092) isin (0 1] such that

[1198791199092]120572(1199092)is nonempty Note that since 119865

119909119860(119905) = sup(119865

119909119910(119905)

119910 isin 119860) for any 119909 isin 119883 and 119905 isin R then 1198651199091[1198791199091]120572(1199091)

(119905) ge

11986511990911199092

(119905) Thus one gets from the contractive condition (12)that

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905)

= min( inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905)

inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)) ge 1198861(119909

1 119909

2 119896119905)

sdot 1198651199091[1198791199091]120572(119909)

(119905) + 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119905)

+ 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905) ge 1198861(119909

1 119909

2 119896119905)

sdot 1198651199091[1198791199091]120572(1199091)

(119905) + 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

+ 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905) = (1198861(119909

1 119909

2 119896119905)

+ 1198863(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905) + 1198862(119909

1 119909

2 119896119905)

sdot 1198651199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

(15)

for any given 120572(1199092) isin (0 1] since 119865

1199091[1198791199091]120572(1199091)

(119905) ge 11986511990911199092

(119905)

for all 119905 isin R+since 119909

2isin [119879119909

1]120572(1199091) Then again since 119909

2isin

[1198791199091]120572(1199091) the following cases can arise for each 119905 isin R

+

4 Journal of Function Spaces

Case (a) One has119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905)

= min( inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905) 1198651199092[1198791199092]120572(1199092)

(119896119905))

le 1198651199092[1198791199092]120572(1199092)

(119896119905)

(16)

for some given 119905 isin R+ Thus from (15) and (16) one gets

1198651199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

(17)

and one gets for the given 119905 isin R+that since 119867(119905) is

nondecreasing and left-continuous then

119865119909[119879119909]

120572(119909)

(119905) ge 119865119909[119879119909]

120572(119909)

(119896119905) forall119905 isin R+ forall119909 isin 119883 (18)

and since 119896 isin (0 1) one gets from (17) that

1198651199092[1198791199092]120572(1199092)

(119905) ge 1198651199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

(19)

and then since [1198791199092]120572(1199092)is closed and nonempty there

exists 1199093isin [119879119909

2]120572(1199092)such that from (19) and the fact that

1198651199092[1198791199092]120572(1199092)

(119905) ge 1198651199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

11986511990921199093

(119905) ge 11986511990921199093

(119896119905) = 1198651199092[1198791199092]120572(1199092)

(119896119905)

ge

1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)

1 minus 1198862(119909

1 119909

2 119896119905)

11986511990911199092

(119905)

forall119905 isin R+

(20)

and equivalently

11986511990921199093

(119905) ge 119892 (1199091 119909

2 119905) 119865

11990911199092

(119896minus1

119905) forall119905 isin R+ (21)

where 119892(1199091 119909

2 119905) = (119886

1(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905))(1 minus 119886

2(119909

1

1199092 119905)) forall119905 isin R

+

Case (b) One has119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)

= min( inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905) 1198651199091[1198791199091]120572(1199091)

(119896119905))

le 1198651199091[1198791199091]120572(1199091)

(119896119905)

(22)

and some 119905 isin R+and 119909

3isin [119879119909

2]120572(1199092)can be chosen for

the previously taken 1199092isin [119879119909

1]120572(1199091)so that 119865

11990921199093

(119896119905) =

inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905) Thus

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)

= 11986511990921199093

(119896119905)

(23)

and since 1198651199092[1198791199092]120572(1199092)

(119896119905) ge 11986511990921199093

(119896119905) one gets for the given119905 isin R

+

11986511990921199093

(119905) ge 11986511990921199093

(119896119905)

ge 1198861(119909

1 119909

2 119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198863(119909

1 119909

2 119905) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119905) 119865

11990921199093

(119896119905)

(24)

Then one gets from (24) that

(1 minus 1198862(119909

1 119909

2 119905)) 119865

11990921199093

(119896119905)

ge (1198861(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905)) 119865

11990911199092

(119905)

(25)

which implies (21) So from Cases (a)-(b) for each given119905 isin R

+ and 119909

1isin 119883 there exist 120572(119909

1) 120572(119909

2) isin (0 1] and

points 1199092isin [119879119909

1]120572(1199091)and 119909

2isin [119879119909

2]120572(1199092)in nonempty level

sets [1198791199091]120572(1199091)and [119879119909

2]120572(1199092)such that (22) holds Proceeding

recursively one gets that a sequence 119909119899may be built for any

arbitrary 1199091= 119909 isin 119883 and 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] forall119899 isin Z

+which satisfies the recursion

11986511990921199093

(119905) ge 119892 (1199091 119909

2 119905) 119865

11990911199092

(119896minus1

119905) forall119905 isin R+

11986511990931199094

(119905) ge 119892 (1199092 119909

3 119905) 119865

11990921199093

(119896minus1

119905)

ge 119892 (1199092 119909

3 119905) 119892 (119909

1 119909

2 119896

minus1

119905) 11986511990911199092

(119896minus2

119905)

forall119905 isin R+

119865119909119899+1119909119899+2

(119905) ge [

119899

prod

119894=1

119892 (119909119894 119909

119894+1 119896

minus119894+1

119905)] 11986511990911199092

(119896minus119899

119905)

forall119905 isin R+ forall119899 isin Z

+

(26)

where

0 lt 119892 (119909119894 119909

119894+1 119905) =

1198861(119909

119894 119909

119894+1 119905) + 119886

3(119909

119894 119909

119894+1 119905)

1 minus 1198862(119909

119894 119909

119894+1 119905)

le 1 forall119905 isin R+ forall119894 isin Z

+

(27)

Note that since0 lt 119886 (119909

119894 119909

119894+1 119905) le 1

0 le 1198862(119909

119894 119909

119894+1 119905) lt 1

forall119905 isin R+ forall119894 isin Z

+

(28)

119899

prod

119894=0

[

1198861(119909

119894 119909

119894+1 119896

minus119894+1

119905) + 1198863(119909

119894 119909

119894+1 119896

minus119894+1

119905)

1 minus 1198862(119909

119894 119909

119894+1 119896

minus119894+1119905)

]

997888rarr 1 as 119899 997888rarr infin forall119905 isin R+

(29)

Journal of Function Spaces 5

then 120575(Γ1119905)120575(Γ

0119905) = +infin forall119905 isin R

+ forall119894 isin Z

+ where 120575(Γ

0119905) =

prodinfin

119895=0[1205750

119895119905] = prod

infin

119895=0[1 minus 120575

1

119895119905] and 120575(Γ

1119905) = prod

infin

119895=0[1205751

119895119905] = prod

infin

119895=0[1 minus

1205750

119895119905] forall119905 isin R

+are discrete measures of the subsequent sets

Γ0119905= Γ

0119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) isin [0 1)

forall119905 isin R+

Γ1119905= Γ

1119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) = 1

forall119905 isin R+

(30)

where 1205750119895119905and 1205751

119895119905are Dirac measures defined by

1205750

119895119905= 1 minus 120575

1

119895119905=

1 if 119895 isin Γ0119905

0 if 119895 notin Γ0119905

forall119905 isin R+

1205751

119895119905= 1 minus 120575

0

119895119905=

1 if 119895 isin Γ1119905

0 if 119895 notin Γ1119905

forall119905 isin R+

(31)

Then one gets from (26) (29) and lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) =

lim120591rarr+infin

minus119867(120591) = 119867(+infinminus

) = 1 forall119905 isin R+ since 119896 isin (0 1)

that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+with 119909

119899+1isin [119879119909

119899]120572(119909119899)

forall119899 isin Z+since

lim119899rarrinfin

119865119909119899+1119909119899+2

(119905) ge ( lim119899rarrinfin

[

119899

prod

119894=1

119892 (119909119894 119909

119894+1 119896

minus119894+1

119905)])

sdot ( lim119899rarrinfin

11986511990911199092

(119896minus119899

119905)) forall119905 isin R+

(32)

Since lim119899rarrinfin

119865119909119899+1119909119899

(119905) = lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) = 1 forall119905 isin

R+and any given 119909

1isin 119883 then for any given 120576 isin R

+

and 120582 isin (0 1) there is 119873 = 119873(120576 120582) isin Z0+

such that119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+ Assume on the contrary

that there exist 119899119896

ge 119873 119899119896+2

gt 119899119896+1

gt 119899119896such that

119865119909119899119896+119894+1

119909119899119896+119894

(120576) ge 119865119909119899119896+119894+1

119909119899119896+119894

(1205762) gt 1 minus 120582 for 119894 = 0 1 and1minus120582 ge 119865

119909119899119896+2

119909119899119896

(120576)Then one has the following contradictionfor the subsequence 119909

119899119896

of 119909119899

1 minus 120582 ge 119865119909119899119896+2

119909119899119896

(120576)

ge Δ119872(119865

119909119899119896+2

119909119899119896+1

(

120576

2

) 119865119909119899119896+1

119909119899119896

(

120576

2

))

gt 1 minus 120582

(33)

Then 119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+

so that 119909119899 is

a Cauchy sequence Since (119883 F Δ119872) is complete one gets

119909119899 rarr 119909

lowast and 120572(119909119899) rarr 120572(119909

lowast

)

It is now proved that 119909lowast isin [119879119909lowast

]120572(119909lowast) Assume on the

contrary that 119909lowast notin [119879119909lowast

]120572(119909lowast) that is (119879119909lowast)(119909lowast) lt 120572(119909

lowast

)

and for some given 119910 isin [119879119909lowast

]120572(119909lowast) there is 119905

1= 119905

1(119910) isin R

+

such that 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) lt 1 for 119905 isin [0 1199051] Then since

119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+and since 119910 = 119909

lowast

1 gt 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119909lowast119910(119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

) 119865119909119899+1119910(

119905

4

)))

= Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

119905

4

) 119865[119879119909119899]120572(119909119899)

119910(

119905

4

)))

119905 isin [0 1199051]

(34)

If 119910 isin [119879119909lowast

]120572(119909lowast)is chosen to fulfill 119865

[119879119909119899]120572(119909119899)

119910(11990514) =

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(11990514) then

1 gt lim sup119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

1199051

2

) Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

119865[119879119909119899]120572(119909119899)

119910(

1199051

4

))) ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

1199051

2

)

Δ119872( lim119899rarrinfin

119865119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

))) = Δ119872(1

Δ119872(1 lim sup

119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)))

= lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)

= lim119899rarrinfin

119865[119879119909lowast]120572(119909lowast)[119879119909lowast]120572(119909lowast)

(

1199051

4

) = 1

for some 1199051isin R

+

(35)

a contradiction Then 119909lowast isin [119879119909lowast]120572(119909lowast)with 120572lowast = 120572(119909

lowast

) thatis it is a probabilistic 120572lowast-fuzzy fixed point of 119879 119883 rarr F(119883)

Corollary 3 Let F(119883) be the collection of all fuzzy sets ina PM-space (119883 F) where 119883 is a nonempty abstract set andlet 119879 119883 rarr F(119883) be a fuzzy mapping Assume that thecontractive condition (12) holds for some real constant 119896 isin

(0 1) subject to condition (2) of Theorem 2 and the specificparticular form of condition (3)

6 Journal of Function Spaces

(31015840

) there exists and strictly increasing sequence of nonnegativeintegers 119873

119899 which satisfies

119873119899+2minus1

prod

119895=119873119899+1

[

1198862(119909 119910 119896

minus119894+1

119905) + 1198863(119909 119910 119896

minus119894+1

119905)

1 minus 1198862(119909 119910 119896

minus119894+1119905)

] =

1

prod119873119899+1minus1

119895=119873119899

[(1198862(119909 119910 119896

minus119894+1119905) + 119886

3(119909 119910 119896

minus119894+1119905)) (1 minus 119886

2(119909 119910 119896

minus119894+1119905))]

forall119905 isin R+ forall119909 isin 119883 119910 isin [119879119909]

120572(119909)sub 119862119861 (119883) forall119899 isin Z

0+

(36)

Proof It is direct from that of Theorem 2 since condition(31015840) guarantees condition (3) of Theorem 2 for any finite firstelement 119873

0isin Z

0+of a strictly increasing sequence 119873

119899

subject to 0 lt 1198721le 119873

119899+1minus 119873

119899le 119872

2lt +infin

Corollary 4 Theorem 2 and Corollary 3 also hold if thecontractive condition (1) is modified as follows

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905)

ge 1198861(119909

1 119909

2 119896119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) + 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905)

= (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

forall119905 isin R+ forall119909

1= 119909 isin 119883 119909

119899+1isin [119879119909

119899]120572(119909119899)sub 119862119861 (119883) forall119899 isin Z

0+

(37)

Proof It is direct from that of Theorem 2 since the abovecontraction condition follows from (12) as a lower-boundwhich has been used in the proof of Theorem 2

Corollary 5 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of119883 and

each sequence 119909119899 sub 119883 of the form 119909

1isin 119883 119909

119899+1isin

[119879119909119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+which satisfies for some

real constant 119896 isin (0 1) the contractive constraint

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge min (1 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905) + 1198863(119909 119910 119896119905) 119865

119909119910(119905))

(38)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894 119883 times 119883 times R

0+rarr [0 1] for

119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Then a sequence 119909119899 may be built for any given arbitrary

1199091= 119909 isin 119883 satisfying 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] and lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isin R+

If in addition (119883 F) is endowed with the minimumtriangular norm Δ

119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each above sequence 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Define the indicator function sequence 120590119894 119883 times 119883 times

R0+

rarr 0 1 119894 isin Z+as

120590119894(119909

119894 119909

119894+1 119896

minus119894+1

119905)

=

1 if 119892 (119909119894 119909

119894+1 119896

minus119894+1

119905) 119865119909119894119909119894+1

(119896minus119894

119905) le 1

0 otherwise

forall119905 isin R+

(39)

with 119892(119909119894 119909

119894+1 119905) defined as in Theorem 2 Then even if for

some 119895 isin Z+and all 119894(ge 119895) isin Z

+ 120590

119894(119909

119894 119909

119894+1 119905) = 0 because

119892(119909119894 119909

119894+1 119896

minus119894+1

119905)119865119909119894119909119894+1

(119896minus119894

119905) gt 1 forall119894(ge 119895) isin Z+ it follows

from (38)-(39) and (26) that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

with 119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+ The rest of the proof is close

to that of Theorem 2

Example 6 It is claimed through this example to revisitthe idea of fuzzy fixed point addressed in [4ndash6 32] tothat of probabilistic fuzzy fixed point according to thedefinition and the formalism given above Consider 119883 =

06 07 08 06 07 08 and let 119879 119883 rarr F(119883) be aprobabilistic fuzzy mapping defined as follows

119879 (06) (119905) = 119879 (08) (119905) =

3

4

if 119905 = 08

1

2

if 119905 = 07

0 if 119905 = 06

119879 (07) (119905) =

0 if 119905 = 08

1

3

if 119905 = 07

3

4

if 119905 = 06

(40)

The 120572-level sets are

[11987906]12

= [11987908]12

= 07 08

[11987906]34

= [11987908]34

= 08

[11987907]34

= 06

[11987907]13

= 07

(41)

Journal of Function Spaces 7

Note that 119865119909[119879119909]

120572(119909)

(119905) = sup(119865119909119910(119905) 119910 isin [119879119909]

120572(119909)) for any

119909 isin 119883 and 119905 isin R and some 120572(119909) isin (0 1] The probabilitydensity functions are as follows with 119865

119909119910(119905) = 119905(119905 + |119909 minus 119910|)

forall119909 119910 isin 119883 forall119905 isin R+

11986506[11987906]

34

(119905) = 1198650608

(119905) =

119905

119905 + 02

forall119905 isin R+

11986506[11987906]

12

(119905) = 1198650607

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

12

(119905) = 1198650807

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

34

(119905) = 1198650808

(119905) = 1 forall119905 isin R+

11986507[11987907]

34

(119905) = 1198650706

(119905) =

119905

119905 + 01

forall119905 isin R+

11986507[11987907]

13

(119905) = 1198650707

(119905) = 1 forall119905 isin R+

119867[119879119909]12[119879119909]12

(119905) =

119905

119905 + 01

for 119909 = 06 08 forall119905 isin R+

119867[119879119909]34[119879119909]34

(119905) = 1 for 119909 = 06 08 forall119905 isin R+

119867[119879119909]12[119879119910]34

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

13

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

34

(119905) =

119905

119905 + 02

for 119909 119910 = 06 08 forall119905 isin R+

119867[11987907]

13[11987907]

13

(119905) = 1 forall119905 isin R+

119867[11987907]

34[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

119867[11987907]

13[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

(42)

Assume that the contractive condition of Theorem 2 holdsunder the form

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905) ge 120572 (119909119899 119909

119899+1 119905 119899)

sdot [119865119909119899[119879119909119899]120572(119909119899)

(119896minus1

119905) + 119865119909119899+1[119879119909119899+1]120572(119909119899+1)

(119896minus1

119905)

+ 119865119909119899119909119899+1

(119896minus1

119905)] 119899 isin Z0+

(43)

for sequences 119909119899 sub 119883 with initial points 119909

0= 119909 isin 119883 119909

1isin

[1198791199090]120572 where 120572(119909 119910 119905 119899) = 120572

119894(119909 119910 119905 119899) 119894 = 1 2 3 forall119905 isin R

+

satisfies

120572 (119909 119910 119905 119899) = 120572119886(119905) le

119896119905

3 (119896119905 + 02)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

120572 (119909 119910 119905 119899) = 120572119887(119905) le

1

3

(1 minus 119890minus120582119905

)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

(44)

for some ℓ(ge 2) isin Z0+ some 119896 isin (0 1) and some 120582 isin R

+

Note that [11987907]13

= 07 with 11986507[11987907]

13

(119905) = 1198650707

(119905) =

1 forall119905 isin R+is a probabilistic 13-fuzzy fixed point of 119879

119883 rarr F(119883) to which the sequences 06 08 07 07 08 06 07 07 converge

3 Further Results

Note that the uniqueness of probabilistic fuzzy fixed points isnot an interesting property to study in the context of prob-abilistic fuzzy fixed point since distinct level sets associatedwith mappings of the form 119879 119883 rarr F(119883) can easily haveintersections of cardinal greater than one in many problemsin the fuzzy context The subsequent result gives conditionsfor the case when several distinct probabilistic fuzzy fixedpoints if they exist are in the intersections of their respectivelevel sets under slightly extended contractive conditions ofthose given in Section 2The extended conditions are of directapplicability

Theorem 7 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 with an

extended contractive condition (12) such that it also holds forany 119909 119910 isin 119883 being probabilistic 120572(119909lowast)-fuzzy fixed points119909 = 119909

lowast

isin [119879119909lowast

]120572(119909lowast)and 119910 = 119910

lowast

isin [119879119910lowast

]120572(119910lowast)of 119879 119883 rarr

F(119883) Then the set [119879119910lowast]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)is nonempty and

119909lowast

119910lowast

isin ([119879119910lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast))

Proof If119909lowast = 119910lowast the result is obvious Assume that there exista probabilistic 120572(119909lowast)-fuzzy fixed point 119909lowast isin [119879119909lowast]

120572(119909lowast)and a

probabilistic 120572(119910lowast)-fuzzy fixed point 119910lowast( = 119909lowast

) isin [119879119910lowast

]120572(119910lowast)

Consider two convergent sequences 119909119899 rarr 119909

lowast and 119910119899 rarr

119910lowast in119883 Then

119865119909119899119910119899

(119905) ge Δ119872(119865

119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

119865119910119899119910lowast (

119905

4

))) forall119905 isin R+

lim inf119899rarrinfin

119865119909119899119910119899

(119905) ge lim inf119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909lowast119910lowast (

119905

4

) 119865119910119899119910lowast (

119905

4

)))

ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

lim119899rarrinfin

119865119910119899119910lowast (

119905

4

))) = Δ119872(1 Δ

119872(119865

119909lowast119910lowast (

119905

4

) 1))

ge 119865119909lowast119910lowast (

119905

4

) forall119905 isin R+

(45)

Assume that 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+

Then from the extended contractive condition (12) and since1 = 119865

119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905) ge 119865119909lowast119910lowast(119905) forall119905 isin R

+

one gets for some 119911119909isin [119879119909

lowast

]120572(119909lowast)and 119911

119910isin [119879119910

lowast

]120572(119910lowast)

8 Journal of Function Spaces

since [119879119909lowast]120572(119909lowast)and [119879119910lowast]

120572(119910lowast)are members of 119862119861(119883) that

is nonempty closed and bounded sets

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905)

= min (119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905))

= min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge 1198861(119909

lowast

119910lowast

119896119905)

sdot 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) + 1198862(119909

lowast

119910lowast

119896119905) 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905)

+ 1198863(119909

lowast

119910lowast

119896119905) 119865119909lowast119910lowast (119905) ge (119886

1(119909

lowast

119910lowast

119896119905)

+ 1198862(119909

lowast

119910lowast

119896119905) + 1198863(119909

lowast

119910lowast

119896119905)) 119865119909lowast119910lowast (119905)

(46)

so that

min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) 119865119909lowast119910lowast (119896

minus1

119905)

ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) Δ119872(119865

119909lowast119911119910

(

119896minus1

119905

2

) 119865119911119910119910lowast (

119896minus1

119905

2

))

ge

119898

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)] Δ119872(119865

119909lowast119911119910

((2119896)minus119898

119905) 119865119911119910119910lowast ((2119896)

119898

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot lim inf119899rarrinfin

Δ119872(119865

119909lowast119911119910

((2119896)minus119899

119905) 119865119911119910119910lowast ((2119896)

minus119899

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot Δ119872(119865

119909lowast119911119910

( lim119899rarrinfin

(2119896)minus119899

119905) 119865119911119910119910lowast ( lim

119899rarrinfin

(2119896)minus119899

119905))

= lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])119867 (+infinminus

) = 1 forall119905 isin R+ forall119898 isin Z

+

(47)

and then 119910lowast = 119911119909 119909lowast = 119911

119910([119879119910

lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)) so that

[119879119910lowast

]120572(119910lowast)cap[119879119909

lowast

]120572(119909lowast)is nonemptyWe can now assume that

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) le 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+ It is direct to

prove in a close way to the above proof that 119910lowast isin [119879119910lowast]120572(119910lowast)cap

[119879119909lowast

]120572(119909lowast)

Remark 8 Note that a direct consequence of Theorem 7from the definition of the level sets is that

119909lowast

119910lowast

isin (( ⋂

120573isin(0120572(119909lowast)]

[119879119909lowast

]120573) cap ( ⋂

120573isin(0120572(119910lowast)]

[119879119910lowast

]120573))

(48)

if 119909lowast isin [119879119909lowast

]120572(119909lowast)and 119910lowast isin [119879119910

lowast

]120572(119910lowast)are any probabilistic

120572(119909lowast

) and 120572(119910lowast)-fuzzy fixed points

Remark 9 Note that corollaries to Theorem 7 can also bestated ldquomutatis-mutandisrdquo under extendedcontractiveconditions

for probabilistic fuzzy fixed points to those given in Corollar-ies 3ndash5

Theorem 10 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 and condi-

tions (1)ndash(3) where the contractive condition

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(49)

is extended to hold for any 119909 119910 isin 119883 with 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Proof Note from Theorem 2 a sequence 119909119899 sub 119883 can be

built being a (convergent) Cauchy sequence such that 119909119899 rarr

Journal of Function Spaces 9

119909lowast 119909

119899+1= 119879119909

119899isin [119879119909

119899]120572(119909119899) forall119899 isin Z

0+for any given 119909

0isin 119883

Then

lim119899rarrinfin

119865119909119899119879119909119899

(119905) = lim119899rarrinfin

119865119909119899119909lowast (119905) = 1 forall119905 isin R

+

lim119899rarrinfin

119865119879119909119899119909lowast (119905) = 1 forall119905 isin R

+

since 119865119879119909119899119909lowast (119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

)) 119865119879119909119899119879119909119899

(

119905

4

))

gt min (1 minus 120582 1 minus 120582 1) = 1 minus 120582

(50)

for any given 119905 isin R+ 120582 isin (0 1) 119899(ge 119873

0) isin Z

0+ and 119873

0=

1198730(120576 120582) ge max(119873

1 119873

2) such that

119865119909lowast119909119899

(

119905

2

) gt 1 minus 120582

for 119899 (ge 1198731) isin Z

0+and some 119873

1= 119873

1(120576 120582) isin Z

0+

119865119909119899+1119909119899

(

119905

4

) gt 1 minus 120582

for 119899 (ge 1198732) isin Z

0+and some 119873

2= 119873

2(120576 120582) isin Z

0+

(51)

since 119865119879119909119899119879119909119899

(1199054) = 1 forall119905 isin R+ forall119899 isin Z

0+from property (1)

of (3) for PM-spaces

Theorem 11 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

(52)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894isin [0 1] for 119894 =

1 3 4 5 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 =

5

sum

119894=1

119886119894le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(53)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(54)

Then a sequence 119909119899 may be constructed for any given

arbitrary 1199091= 119909 isin 119883 such that 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883)

with 120572(119909119899) sube (0 1] satisfying lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isinR+If in addition (119883 F) is endowed with the minimum

triangular norm Δ119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Since [119879119909]120572(119909)

[119879119910]120572(119910)

sub 119883

max (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min119911isin[119879119909]

120572(119909)120596isin[119879119910]

120572(119910)

(119865119911[119879119910]

120572(119910)

(119896119905) 119865120596[119879119909]

120572(119909)

(119896119905))

= 119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905) + 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

forall119905 isin R+ forall119909 119910 isin 119883

(55)

Now for any given 119909 119910 isin 119883 Then the following cases canoccur

(a) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905) forall119905 isin R+

(1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4) 119865

119910[119879119910]120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4+ 119886

5) 119865

119909119910(119905) forall119905 isin R

+

(56)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(57)

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of Function Spaces 3

For 119860 119861 isin F(119883) 119860 sub 119861 means 119860(119909) le 119861(119909) forall119909 isin 119883Note also that if 120572 isin [120573 1] and 120573 isin (0 1] then [119860]

120572sube [119860]

120573

If there exists 120572 isin [0 1] such that [119860]120572 [119861]

120572isin 119862119861(119883) then

define

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905)

= min( inf119886isin[119879119909]

119865119886[119879119910]

120572(119910)

(119905) inf119887isin[119879119910]

119865119887[119879119909]

120572(119909)

(119905))

forall119905 isin R

(11)

The collection of all the approximate quantities in a metriclinear space 119883 is denoted by119882(119883) 119879 119883 rarr F(119884) is a fuzzymapping from an arbitrary set 119883 to F(119884) which is a fuzzysubset in119883times119884 and the grade of membership of 119910 in 119879(119909) is119879(119909)(119910)

The notation 119891 119883 | 119884 rarr 119885 means that the domain ofthe function 119891 from119883 to 119885 is restricted to the subset 119884 of119883

The next definition characterizes probabilistic fuzzy fixedpoints in an appropriate way to establish some results of thispaper

Definition 1 If F(119883) is the collection of all fuzzy sets in thePM-space (119883 F) where119883 is a nonempty abstract set and 119879

119883 rarr F(119883) is a fuzzy mapping then 119909 isin 119883 is a probabilistic120572-fuzzy fixed point of 119879 119883 rarr F(119883) if for some 120572 isin (0 1][119879119909]

120572isin 119862119861(119883) and 119909 isin [119879119909]

120572 that is (119879119909)(119909) ge 120572

Note that if 119883 is a nonempty abstract set (119883 F) is a PM-space 119860 isin F(119883) and for some 120572 isin (0 1] [119860]

120572isin 119862119861(119883)

119879 119883 rarr F(119883) then

(1) 119865[119860]120572[119860]120572

(119905) = 1 forall119905 isin R+

(2) if 119909 isin 119883 is a probabilistic 120572-fuzzy fixed point of 119879

119883 rarr F(119883) then 119865119909[119879119909]

120572

(119905) = 1 forall119905 isin R+

(3) if [119879119909]120572(119909)

isin 119862119861(119883) and 119909 isin 119883 is not a probabilistic120572-fuzzy fixed point of 119879 119883 rarr F(119883) then 119909 notin

[119879119909]120572(119909)

equivalently (119879119909)(119909) lt 120572 and 119865119909[119879119909]

120572

(119905) lt

1 119905 isin [0 1199051] for some 119905

1= 119905

1(119909) isin R

+

The following result holds

Theorem 2 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions hold

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(12)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 (119909 119910 119905) =

3

sum

119894=1

119886119894(119909 119910 119905) le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(13)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(14)

Then a sequence 119909119899may be built for any given arbitrary 119909

1=

119909 isin 119883 satisfying 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) with 120572(119909

119899) sube

(0 1] satisfying lim119899rarrinfin

119865119909119899119909119899+1

(119905) = 1 forall119905 isin R+

If in addition (119883 F) is endowed with the minimumtriangular norm Δ

119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Take arbitrary points 1199091= 119909 isin 119883 119909

2= 119910 isin [119879119909

1]120572(1199091)

for some given existing 120572(1199091) isin (0 1] such that [119879119909

1]120572(1199091)is

nonempty and take also some existing120572(1199092) isin (0 1] such that

[1198791199092]120572(1199092)is nonempty Note that since 119865

119909119860(119905) = sup(119865

119909119910(119905)

119910 isin 119860) for any 119909 isin 119883 and 119905 isin R then 1198651199091[1198791199091]120572(1199091)

(119905) ge

11986511990911199092

(119905) Thus one gets from the contractive condition (12)that

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905)

= min( inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905)

inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)) ge 1198861(119909

1 119909

2 119896119905)

sdot 1198651199091[1198791199091]120572(119909)

(119905) + 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119905)

+ 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905) ge 1198861(119909

1 119909

2 119896119905)

sdot 1198651199091[1198791199091]120572(1199091)

(119905) + 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

+ 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905) = (1198861(119909

1 119909

2 119896119905)

+ 1198863(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905) + 1198862(119909

1 119909

2 119896119905)

sdot 1198651199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

(15)

for any given 120572(1199092) isin (0 1] since 119865

1199091[1198791199091]120572(1199091)

(119905) ge 11986511990911199092

(119905)

for all 119905 isin R+since 119909

2isin [119879119909

1]120572(1199091) Then again since 119909

2isin

[1198791199091]120572(1199091) the following cases can arise for each 119905 isin R

+

4 Journal of Function Spaces

Case (a) One has119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905)

= min( inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905) 1198651199092[1198791199092]120572(1199092)

(119896119905))

le 1198651199092[1198791199092]120572(1199092)

(119896119905)

(16)

for some given 119905 isin R+ Thus from (15) and (16) one gets

1198651199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

(17)

and one gets for the given 119905 isin R+that since 119867(119905) is

nondecreasing and left-continuous then

119865119909[119879119909]

120572(119909)

(119905) ge 119865119909[119879119909]

120572(119909)

(119896119905) forall119905 isin R+ forall119909 isin 119883 (18)

and since 119896 isin (0 1) one gets from (17) that

1198651199092[1198791199092]120572(1199092)

(119905) ge 1198651199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

(19)

and then since [1198791199092]120572(1199092)is closed and nonempty there

exists 1199093isin [119879119909

2]120572(1199092)such that from (19) and the fact that

1198651199092[1198791199092]120572(1199092)

(119905) ge 1198651199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

11986511990921199093

(119905) ge 11986511990921199093

(119896119905) = 1198651199092[1198791199092]120572(1199092)

(119896119905)

ge

1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)

1 minus 1198862(119909

1 119909

2 119896119905)

11986511990911199092

(119905)

forall119905 isin R+

(20)

and equivalently

11986511990921199093

(119905) ge 119892 (1199091 119909

2 119905) 119865

11990911199092

(119896minus1

119905) forall119905 isin R+ (21)

where 119892(1199091 119909

2 119905) = (119886

1(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905))(1 minus 119886

2(119909

1

1199092 119905)) forall119905 isin R

+

Case (b) One has119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)

= min( inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905) 1198651199091[1198791199091]120572(1199091)

(119896119905))

le 1198651199091[1198791199091]120572(1199091)

(119896119905)

(22)

and some 119905 isin R+and 119909

3isin [119879119909

2]120572(1199092)can be chosen for

the previously taken 1199092isin [119879119909

1]120572(1199091)so that 119865

11990921199093

(119896119905) =

inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905) Thus

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)

= 11986511990921199093

(119896119905)

(23)

and since 1198651199092[1198791199092]120572(1199092)

(119896119905) ge 11986511990921199093

(119896119905) one gets for the given119905 isin R

+

11986511990921199093

(119905) ge 11986511990921199093

(119896119905)

ge 1198861(119909

1 119909

2 119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198863(119909

1 119909

2 119905) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119905) 119865

11990921199093

(119896119905)

(24)

Then one gets from (24) that

(1 minus 1198862(119909

1 119909

2 119905)) 119865

11990921199093

(119896119905)

ge (1198861(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905)) 119865

11990911199092

(119905)

(25)

which implies (21) So from Cases (a)-(b) for each given119905 isin R

+ and 119909

1isin 119883 there exist 120572(119909

1) 120572(119909

2) isin (0 1] and

points 1199092isin [119879119909

1]120572(1199091)and 119909

2isin [119879119909

2]120572(1199092)in nonempty level

sets [1198791199091]120572(1199091)and [119879119909

2]120572(1199092)such that (22) holds Proceeding

recursively one gets that a sequence 119909119899may be built for any

arbitrary 1199091= 119909 isin 119883 and 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] forall119899 isin Z

+which satisfies the recursion

11986511990921199093

(119905) ge 119892 (1199091 119909

2 119905) 119865

11990911199092

(119896minus1

119905) forall119905 isin R+

11986511990931199094

(119905) ge 119892 (1199092 119909

3 119905) 119865

11990921199093

(119896minus1

119905)

ge 119892 (1199092 119909

3 119905) 119892 (119909

1 119909

2 119896

minus1

119905) 11986511990911199092

(119896minus2

119905)

forall119905 isin R+

119865119909119899+1119909119899+2

(119905) ge [

119899

prod

119894=1

119892 (119909119894 119909

119894+1 119896

minus119894+1

119905)] 11986511990911199092

(119896minus119899

119905)

forall119905 isin R+ forall119899 isin Z

+

(26)

where

0 lt 119892 (119909119894 119909

119894+1 119905) =

1198861(119909

119894 119909

119894+1 119905) + 119886

3(119909

119894 119909

119894+1 119905)

1 minus 1198862(119909

119894 119909

119894+1 119905)

le 1 forall119905 isin R+ forall119894 isin Z

+

(27)

Note that since0 lt 119886 (119909

119894 119909

119894+1 119905) le 1

0 le 1198862(119909

119894 119909

119894+1 119905) lt 1

forall119905 isin R+ forall119894 isin Z

+

(28)

119899

prod

119894=0

[

1198861(119909

119894 119909

119894+1 119896

minus119894+1

119905) + 1198863(119909

119894 119909

119894+1 119896

minus119894+1

119905)

1 minus 1198862(119909

119894 119909

119894+1 119896

minus119894+1119905)

]

997888rarr 1 as 119899 997888rarr infin forall119905 isin R+

(29)

Journal of Function Spaces 5

then 120575(Γ1119905)120575(Γ

0119905) = +infin forall119905 isin R

+ forall119894 isin Z

+ where 120575(Γ

0119905) =

prodinfin

119895=0[1205750

119895119905] = prod

infin

119895=0[1 minus 120575

1

119895119905] and 120575(Γ

1119905) = prod

infin

119895=0[1205751

119895119905] = prod

infin

119895=0[1 minus

1205750

119895119905] forall119905 isin R

+are discrete measures of the subsequent sets

Γ0119905= Γ

0119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) isin [0 1)

forall119905 isin R+

Γ1119905= Γ

1119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) = 1

forall119905 isin R+

(30)

where 1205750119895119905and 1205751

119895119905are Dirac measures defined by

1205750

119895119905= 1 minus 120575

1

119895119905=

1 if 119895 isin Γ0119905

0 if 119895 notin Γ0119905

forall119905 isin R+

1205751

119895119905= 1 minus 120575

0

119895119905=

1 if 119895 isin Γ1119905

0 if 119895 notin Γ1119905

forall119905 isin R+

(31)

Then one gets from (26) (29) and lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) =

lim120591rarr+infin

minus119867(120591) = 119867(+infinminus

) = 1 forall119905 isin R+ since 119896 isin (0 1)

that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+with 119909

119899+1isin [119879119909

119899]120572(119909119899)

forall119899 isin Z+since

lim119899rarrinfin

119865119909119899+1119909119899+2

(119905) ge ( lim119899rarrinfin

[

119899

prod

119894=1

119892 (119909119894 119909

119894+1 119896

minus119894+1

119905)])

sdot ( lim119899rarrinfin

11986511990911199092

(119896minus119899

119905)) forall119905 isin R+

(32)

Since lim119899rarrinfin

119865119909119899+1119909119899

(119905) = lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) = 1 forall119905 isin

R+and any given 119909

1isin 119883 then for any given 120576 isin R

+

and 120582 isin (0 1) there is 119873 = 119873(120576 120582) isin Z0+

such that119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+ Assume on the contrary

that there exist 119899119896

ge 119873 119899119896+2

gt 119899119896+1

gt 119899119896such that

119865119909119899119896+119894+1

119909119899119896+119894

(120576) ge 119865119909119899119896+119894+1

119909119899119896+119894

(1205762) gt 1 minus 120582 for 119894 = 0 1 and1minus120582 ge 119865

119909119899119896+2

119909119899119896

(120576)Then one has the following contradictionfor the subsequence 119909

119899119896

of 119909119899

1 minus 120582 ge 119865119909119899119896+2

119909119899119896

(120576)

ge Δ119872(119865

119909119899119896+2

119909119899119896+1

(

120576

2

) 119865119909119899119896+1

119909119899119896

(

120576

2

))

gt 1 minus 120582

(33)

Then 119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+

so that 119909119899 is

a Cauchy sequence Since (119883 F Δ119872) is complete one gets

119909119899 rarr 119909

lowast and 120572(119909119899) rarr 120572(119909

lowast

)

It is now proved that 119909lowast isin [119879119909lowast

]120572(119909lowast) Assume on the

contrary that 119909lowast notin [119879119909lowast

]120572(119909lowast) that is (119879119909lowast)(119909lowast) lt 120572(119909

lowast

)

and for some given 119910 isin [119879119909lowast

]120572(119909lowast) there is 119905

1= 119905

1(119910) isin R

+

such that 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) lt 1 for 119905 isin [0 1199051] Then since

119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+and since 119910 = 119909

lowast

1 gt 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119909lowast119910(119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

) 119865119909119899+1119910(

119905

4

)))

= Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

119905

4

) 119865[119879119909119899]120572(119909119899)

119910(

119905

4

)))

119905 isin [0 1199051]

(34)

If 119910 isin [119879119909lowast

]120572(119909lowast)is chosen to fulfill 119865

[119879119909119899]120572(119909119899)

119910(11990514) =

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(11990514) then

1 gt lim sup119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

1199051

2

) Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

119865[119879119909119899]120572(119909119899)

119910(

1199051

4

))) ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

1199051

2

)

Δ119872( lim119899rarrinfin

119865119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

))) = Δ119872(1

Δ119872(1 lim sup

119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)))

= lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)

= lim119899rarrinfin

119865[119879119909lowast]120572(119909lowast)[119879119909lowast]120572(119909lowast)

(

1199051

4

) = 1

for some 1199051isin R

+

(35)

a contradiction Then 119909lowast isin [119879119909lowast]120572(119909lowast)with 120572lowast = 120572(119909

lowast

) thatis it is a probabilistic 120572lowast-fuzzy fixed point of 119879 119883 rarr F(119883)

Corollary 3 Let F(119883) be the collection of all fuzzy sets ina PM-space (119883 F) where 119883 is a nonempty abstract set andlet 119879 119883 rarr F(119883) be a fuzzy mapping Assume that thecontractive condition (12) holds for some real constant 119896 isin

(0 1) subject to condition (2) of Theorem 2 and the specificparticular form of condition (3)

6 Journal of Function Spaces

(31015840

) there exists and strictly increasing sequence of nonnegativeintegers 119873

119899 which satisfies

119873119899+2minus1

prod

119895=119873119899+1

[

1198862(119909 119910 119896

minus119894+1

119905) + 1198863(119909 119910 119896

minus119894+1

119905)

1 minus 1198862(119909 119910 119896

minus119894+1119905)

] =

1

prod119873119899+1minus1

119895=119873119899

[(1198862(119909 119910 119896

minus119894+1119905) + 119886

3(119909 119910 119896

minus119894+1119905)) (1 minus 119886

2(119909 119910 119896

minus119894+1119905))]

forall119905 isin R+ forall119909 isin 119883 119910 isin [119879119909]

120572(119909)sub 119862119861 (119883) forall119899 isin Z

0+

(36)

Proof It is direct from that of Theorem 2 since condition(31015840) guarantees condition (3) of Theorem 2 for any finite firstelement 119873

0isin Z

0+of a strictly increasing sequence 119873

119899

subject to 0 lt 1198721le 119873

119899+1minus 119873

119899le 119872

2lt +infin

Corollary 4 Theorem 2 and Corollary 3 also hold if thecontractive condition (1) is modified as follows

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905)

ge 1198861(119909

1 119909

2 119896119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) + 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905)

= (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

forall119905 isin R+ forall119909

1= 119909 isin 119883 119909

119899+1isin [119879119909

119899]120572(119909119899)sub 119862119861 (119883) forall119899 isin Z

0+

(37)

Proof It is direct from that of Theorem 2 since the abovecontraction condition follows from (12) as a lower-boundwhich has been used in the proof of Theorem 2

Corollary 5 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of119883 and

each sequence 119909119899 sub 119883 of the form 119909

1isin 119883 119909

119899+1isin

[119879119909119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+which satisfies for some

real constant 119896 isin (0 1) the contractive constraint

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge min (1 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905) + 1198863(119909 119910 119896119905) 119865

119909119910(119905))

(38)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894 119883 times 119883 times R

0+rarr [0 1] for

119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Then a sequence 119909119899 may be built for any given arbitrary

1199091= 119909 isin 119883 satisfying 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] and lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isin R+

If in addition (119883 F) is endowed with the minimumtriangular norm Δ

119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each above sequence 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Define the indicator function sequence 120590119894 119883 times 119883 times

R0+

rarr 0 1 119894 isin Z+as

120590119894(119909

119894 119909

119894+1 119896

minus119894+1

119905)

=

1 if 119892 (119909119894 119909

119894+1 119896

minus119894+1

119905) 119865119909119894119909119894+1

(119896minus119894

119905) le 1

0 otherwise

forall119905 isin R+

(39)

with 119892(119909119894 119909

119894+1 119905) defined as in Theorem 2 Then even if for

some 119895 isin Z+and all 119894(ge 119895) isin Z

+ 120590

119894(119909

119894 119909

119894+1 119905) = 0 because

119892(119909119894 119909

119894+1 119896

minus119894+1

119905)119865119909119894119909119894+1

(119896minus119894

119905) gt 1 forall119894(ge 119895) isin Z+ it follows

from (38)-(39) and (26) that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

with 119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+ The rest of the proof is close

to that of Theorem 2

Example 6 It is claimed through this example to revisitthe idea of fuzzy fixed point addressed in [4ndash6 32] tothat of probabilistic fuzzy fixed point according to thedefinition and the formalism given above Consider 119883 =

06 07 08 06 07 08 and let 119879 119883 rarr F(119883) be aprobabilistic fuzzy mapping defined as follows

119879 (06) (119905) = 119879 (08) (119905) =

3

4

if 119905 = 08

1

2

if 119905 = 07

0 if 119905 = 06

119879 (07) (119905) =

0 if 119905 = 08

1

3

if 119905 = 07

3

4

if 119905 = 06

(40)

The 120572-level sets are

[11987906]12

= [11987908]12

= 07 08

[11987906]34

= [11987908]34

= 08

[11987907]34

= 06

[11987907]13

= 07

(41)

Journal of Function Spaces 7

Note that 119865119909[119879119909]

120572(119909)

(119905) = sup(119865119909119910(119905) 119910 isin [119879119909]

120572(119909)) for any

119909 isin 119883 and 119905 isin R and some 120572(119909) isin (0 1] The probabilitydensity functions are as follows with 119865

119909119910(119905) = 119905(119905 + |119909 minus 119910|)

forall119909 119910 isin 119883 forall119905 isin R+

11986506[11987906]

34

(119905) = 1198650608

(119905) =

119905

119905 + 02

forall119905 isin R+

11986506[11987906]

12

(119905) = 1198650607

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

12

(119905) = 1198650807

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

34

(119905) = 1198650808

(119905) = 1 forall119905 isin R+

11986507[11987907]

34

(119905) = 1198650706

(119905) =

119905

119905 + 01

forall119905 isin R+

11986507[11987907]

13

(119905) = 1198650707

(119905) = 1 forall119905 isin R+

119867[119879119909]12[119879119909]12

(119905) =

119905

119905 + 01

for 119909 = 06 08 forall119905 isin R+

119867[119879119909]34[119879119909]34

(119905) = 1 for 119909 = 06 08 forall119905 isin R+

119867[119879119909]12[119879119910]34

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

13

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

34

(119905) =

119905

119905 + 02

for 119909 119910 = 06 08 forall119905 isin R+

119867[11987907]

13[11987907]

13

(119905) = 1 forall119905 isin R+

119867[11987907]

34[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

119867[11987907]

13[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

(42)

Assume that the contractive condition of Theorem 2 holdsunder the form

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905) ge 120572 (119909119899 119909

119899+1 119905 119899)

sdot [119865119909119899[119879119909119899]120572(119909119899)

(119896minus1

119905) + 119865119909119899+1[119879119909119899+1]120572(119909119899+1)

(119896minus1

119905)

+ 119865119909119899119909119899+1

(119896minus1

119905)] 119899 isin Z0+

(43)

for sequences 119909119899 sub 119883 with initial points 119909

0= 119909 isin 119883 119909

1isin

[1198791199090]120572 where 120572(119909 119910 119905 119899) = 120572

119894(119909 119910 119905 119899) 119894 = 1 2 3 forall119905 isin R

+

satisfies

120572 (119909 119910 119905 119899) = 120572119886(119905) le

119896119905

3 (119896119905 + 02)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

120572 (119909 119910 119905 119899) = 120572119887(119905) le

1

3

(1 minus 119890minus120582119905

)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

(44)

for some ℓ(ge 2) isin Z0+ some 119896 isin (0 1) and some 120582 isin R

+

Note that [11987907]13

= 07 with 11986507[11987907]

13

(119905) = 1198650707

(119905) =

1 forall119905 isin R+is a probabilistic 13-fuzzy fixed point of 119879

119883 rarr F(119883) to which the sequences 06 08 07 07 08 06 07 07 converge

3 Further Results

Note that the uniqueness of probabilistic fuzzy fixed points isnot an interesting property to study in the context of prob-abilistic fuzzy fixed point since distinct level sets associatedwith mappings of the form 119879 119883 rarr F(119883) can easily haveintersections of cardinal greater than one in many problemsin the fuzzy context The subsequent result gives conditionsfor the case when several distinct probabilistic fuzzy fixedpoints if they exist are in the intersections of their respectivelevel sets under slightly extended contractive conditions ofthose given in Section 2The extended conditions are of directapplicability

Theorem 7 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 with an

extended contractive condition (12) such that it also holds forany 119909 119910 isin 119883 being probabilistic 120572(119909lowast)-fuzzy fixed points119909 = 119909

lowast

isin [119879119909lowast

]120572(119909lowast)and 119910 = 119910

lowast

isin [119879119910lowast

]120572(119910lowast)of 119879 119883 rarr

F(119883) Then the set [119879119910lowast]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)is nonempty and

119909lowast

119910lowast

isin ([119879119910lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast))

Proof If119909lowast = 119910lowast the result is obvious Assume that there exista probabilistic 120572(119909lowast)-fuzzy fixed point 119909lowast isin [119879119909lowast]

120572(119909lowast)and a

probabilistic 120572(119910lowast)-fuzzy fixed point 119910lowast( = 119909lowast

) isin [119879119910lowast

]120572(119910lowast)

Consider two convergent sequences 119909119899 rarr 119909

lowast and 119910119899 rarr

119910lowast in119883 Then

119865119909119899119910119899

(119905) ge Δ119872(119865

119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

119865119910119899119910lowast (

119905

4

))) forall119905 isin R+

lim inf119899rarrinfin

119865119909119899119910119899

(119905) ge lim inf119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909lowast119910lowast (

119905

4

) 119865119910119899119910lowast (

119905

4

)))

ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

lim119899rarrinfin

119865119910119899119910lowast (

119905

4

))) = Δ119872(1 Δ

119872(119865

119909lowast119910lowast (

119905

4

) 1))

ge 119865119909lowast119910lowast (

119905

4

) forall119905 isin R+

(45)

Assume that 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+

Then from the extended contractive condition (12) and since1 = 119865

119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905) ge 119865119909lowast119910lowast(119905) forall119905 isin R

+

one gets for some 119911119909isin [119879119909

lowast

]120572(119909lowast)and 119911

119910isin [119879119910

lowast

]120572(119910lowast)

8 Journal of Function Spaces

since [119879119909lowast]120572(119909lowast)and [119879119910lowast]

120572(119910lowast)are members of 119862119861(119883) that

is nonempty closed and bounded sets

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905)

= min (119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905))

= min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge 1198861(119909

lowast

119910lowast

119896119905)

sdot 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) + 1198862(119909

lowast

119910lowast

119896119905) 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905)

+ 1198863(119909

lowast

119910lowast

119896119905) 119865119909lowast119910lowast (119905) ge (119886

1(119909

lowast

119910lowast

119896119905)

+ 1198862(119909

lowast

119910lowast

119896119905) + 1198863(119909

lowast

119910lowast

119896119905)) 119865119909lowast119910lowast (119905)

(46)

so that

min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) 119865119909lowast119910lowast (119896

minus1

119905)

ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) Δ119872(119865

119909lowast119911119910

(

119896minus1

119905

2

) 119865119911119910119910lowast (

119896minus1

119905

2

))

ge

119898

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)] Δ119872(119865

119909lowast119911119910

((2119896)minus119898

119905) 119865119911119910119910lowast ((2119896)

119898

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot lim inf119899rarrinfin

Δ119872(119865

119909lowast119911119910

((2119896)minus119899

119905) 119865119911119910119910lowast ((2119896)

minus119899

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot Δ119872(119865

119909lowast119911119910

( lim119899rarrinfin

(2119896)minus119899

119905) 119865119911119910119910lowast ( lim

119899rarrinfin

(2119896)minus119899

119905))

= lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])119867 (+infinminus

) = 1 forall119905 isin R+ forall119898 isin Z

+

(47)

and then 119910lowast = 119911119909 119909lowast = 119911

119910([119879119910

lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)) so that

[119879119910lowast

]120572(119910lowast)cap[119879119909

lowast

]120572(119909lowast)is nonemptyWe can now assume that

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) le 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+ It is direct to

prove in a close way to the above proof that 119910lowast isin [119879119910lowast]120572(119910lowast)cap

[119879119909lowast

]120572(119909lowast)

Remark 8 Note that a direct consequence of Theorem 7from the definition of the level sets is that

119909lowast

119910lowast

isin (( ⋂

120573isin(0120572(119909lowast)]

[119879119909lowast

]120573) cap ( ⋂

120573isin(0120572(119910lowast)]

[119879119910lowast

]120573))

(48)

if 119909lowast isin [119879119909lowast

]120572(119909lowast)and 119910lowast isin [119879119910

lowast

]120572(119910lowast)are any probabilistic

120572(119909lowast

) and 120572(119910lowast)-fuzzy fixed points

Remark 9 Note that corollaries to Theorem 7 can also bestated ldquomutatis-mutandisrdquo under extendedcontractiveconditions

for probabilistic fuzzy fixed points to those given in Corollar-ies 3ndash5

Theorem 10 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 and condi-

tions (1)ndash(3) where the contractive condition

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(49)

is extended to hold for any 119909 119910 isin 119883 with 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Proof Note from Theorem 2 a sequence 119909119899 sub 119883 can be

built being a (convergent) Cauchy sequence such that 119909119899 rarr

Journal of Function Spaces 9

119909lowast 119909

119899+1= 119879119909

119899isin [119879119909

119899]120572(119909119899) forall119899 isin Z

0+for any given 119909

0isin 119883

Then

lim119899rarrinfin

119865119909119899119879119909119899

(119905) = lim119899rarrinfin

119865119909119899119909lowast (119905) = 1 forall119905 isin R

+

lim119899rarrinfin

119865119879119909119899119909lowast (119905) = 1 forall119905 isin R

+

since 119865119879119909119899119909lowast (119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

)) 119865119879119909119899119879119909119899

(

119905

4

))

gt min (1 minus 120582 1 minus 120582 1) = 1 minus 120582

(50)

for any given 119905 isin R+ 120582 isin (0 1) 119899(ge 119873

0) isin Z

0+ and 119873

0=

1198730(120576 120582) ge max(119873

1 119873

2) such that

119865119909lowast119909119899

(

119905

2

) gt 1 minus 120582

for 119899 (ge 1198731) isin Z

0+and some 119873

1= 119873

1(120576 120582) isin Z

0+

119865119909119899+1119909119899

(

119905

4

) gt 1 minus 120582

for 119899 (ge 1198732) isin Z

0+and some 119873

2= 119873

2(120576 120582) isin Z

0+

(51)

since 119865119879119909119899119879119909119899

(1199054) = 1 forall119905 isin R+ forall119899 isin Z

0+from property (1)

of (3) for PM-spaces

Theorem 11 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

(52)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894isin [0 1] for 119894 =

1 3 4 5 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 =

5

sum

119894=1

119886119894le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(53)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(54)

Then a sequence 119909119899 may be constructed for any given

arbitrary 1199091= 119909 isin 119883 such that 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883)

with 120572(119909119899) sube (0 1] satisfying lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isinR+If in addition (119883 F) is endowed with the minimum

triangular norm Δ119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Since [119879119909]120572(119909)

[119879119910]120572(119910)

sub 119883

max (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min119911isin[119879119909]

120572(119909)120596isin[119879119910]

120572(119910)

(119865119911[119879119910]

120572(119910)

(119896119905) 119865120596[119879119909]

120572(119909)

(119896119905))

= 119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905) + 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

forall119905 isin R+ forall119909 119910 isin 119883

(55)

Now for any given 119909 119910 isin 119883 Then the following cases canoccur

(a) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905) forall119905 isin R+

(1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4) 119865

119910[119879119910]120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4+ 119886

5) 119865

119909119910(119905) forall119905 isin R

+

(56)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(57)

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

Case (a) One has119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905)

= min( inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905) 1198651199092[1198791199092]120572(1199092)

(119896119905))

le 1198651199092[1198791199092]120572(1199092)

(119896119905)

(16)

for some given 119905 isin R+ Thus from (15) and (16) one gets

1198651199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

(17)

and one gets for the given 119905 isin R+that since 119867(119905) is

nondecreasing and left-continuous then

119865119909[119879119909]

120572(119909)

(119905) ge 119865119909[119879119909]

120572(119909)

(119896119905) forall119905 isin R+ forall119909 isin 119883 (18)

and since 119896 isin (0 1) one gets from (17) that

1198651199092[1198791199092]120572(1199092)

(119905) ge 1198651199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

(19)

and then since [1198791199092]120572(1199092)is closed and nonempty there

exists 1199093isin [119879119909

2]120572(1199092)such that from (19) and the fact that

1198651199092[1198791199092]120572(1199092)

(119905) ge 1198651199092[1198791199092]120572(1199092)

(119896119905) forall119905 isin R+

11986511990921199093

(119905) ge 11986511990921199093

(119896119905) = 1198651199092[1198791199092]120572(1199092)

(119896119905)

ge

1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)

1 minus 1198862(119909

1 119909

2 119896119905)

11986511990911199092

(119905)

forall119905 isin R+

(20)

and equivalently

11986511990921199093

(119905) ge 119892 (1199091 119909

2 119905) 119865

11990911199092

(119896minus1

119905) forall119905 isin R+ (21)

where 119892(1199091 119909

2 119905) = (119886

1(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905))(1 minus 119886

2(119909

1

1199092 119905)) forall119905 isin R

+

Case (b) One has119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)

= min( inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905) 1198651199091[1198791199091]120572(1199091)

(119896119905))

le 1198651199091[1198791199091]120572(1199091)

(119896119905)

(22)

and some 119905 isin R+and 119909

3isin [119879119909

2]120572(1199092)can be chosen for

the previously taken 1199092isin [119879119909

1]120572(1199091)so that 119865

11990921199093

(119896119905) =

inf119886isin[119879119909

1]120572(1199091)

119865119886[1198791199092]120572(1199092)

(119896119905) Thus

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905) = inf119887isin[119879119909

2]120572(1199092)

119865119887[1198791199091]120572(1199091)

(119896119905)

= 11986511990921199093

(119896119905)

(23)

and since 1198651199092[1198791199092]120572(1199092)

(119896119905) ge 11986511990921199093

(119896119905) one gets for the given119905 isin R

+

11986511990921199093

(119905) ge 11986511990921199093

(119896119905)

ge 1198861(119909

1 119909

2 119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198863(119909

1 119909

2 119905) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

ge (1198861(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119905) 119865

11990921199093

(119896119905)

(24)

Then one gets from (24) that

(1 minus 1198862(119909

1 119909

2 119905)) 119865

11990921199093

(119896119905)

ge (1198861(119909

1 119909

2 119905) + 119886

3(119909

1 119909

2 119905)) 119865

11990911199092

(119905)

(25)

which implies (21) So from Cases (a)-(b) for each given119905 isin R

+ and 119909

1isin 119883 there exist 120572(119909

1) 120572(119909

2) isin (0 1] and

points 1199092isin [119879119909

1]120572(1199091)and 119909

2isin [119879119909

2]120572(1199092)in nonempty level

sets [1198791199091]120572(1199091)and [119879119909

2]120572(1199092)such that (22) holds Proceeding

recursively one gets that a sequence 119909119899may be built for any

arbitrary 1199091= 119909 isin 119883 and 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] forall119899 isin Z

+which satisfies the recursion

11986511990921199093

(119905) ge 119892 (1199091 119909

2 119905) 119865

11990911199092

(119896minus1

119905) forall119905 isin R+

11986511990931199094

(119905) ge 119892 (1199092 119909

3 119905) 119865

11990921199093

(119896minus1

119905)

ge 119892 (1199092 119909

3 119905) 119892 (119909

1 119909

2 119896

minus1

119905) 11986511990911199092

(119896minus2

119905)

forall119905 isin R+

119865119909119899+1119909119899+2

(119905) ge [

119899

prod

119894=1

119892 (119909119894 119909

119894+1 119896

minus119894+1

119905)] 11986511990911199092

(119896minus119899

119905)

forall119905 isin R+ forall119899 isin Z

+

(26)

where

0 lt 119892 (119909119894 119909

119894+1 119905) =

1198861(119909

119894 119909

119894+1 119905) + 119886

3(119909

119894 119909

119894+1 119905)

1 minus 1198862(119909

119894 119909

119894+1 119905)

le 1 forall119905 isin R+ forall119894 isin Z

+

(27)

Note that since0 lt 119886 (119909

119894 119909

119894+1 119905) le 1

0 le 1198862(119909

119894 119909

119894+1 119905) lt 1

forall119905 isin R+ forall119894 isin Z

+

(28)

119899

prod

119894=0

[

1198861(119909

119894 119909

119894+1 119896

minus119894+1

119905) + 1198863(119909

119894 119909

119894+1 119896

minus119894+1

119905)

1 minus 1198862(119909

119894 119909

119894+1 119896

minus119894+1119905)

]

997888rarr 1 as 119899 997888rarr infin forall119905 isin R+

(29)

Journal of Function Spaces 5

then 120575(Γ1119905)120575(Γ

0119905) = +infin forall119905 isin R

+ forall119894 isin Z

+ where 120575(Γ

0119905) =

prodinfin

119895=0[1205750

119895119905] = prod

infin

119895=0[1 minus 120575

1

119895119905] and 120575(Γ

1119905) = prod

infin

119895=0[1205751

119895119905] = prod

infin

119895=0[1 minus

1205750

119895119905] forall119905 isin R

+are discrete measures of the subsequent sets

Γ0119905= Γ

0119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) isin [0 1)

forall119905 isin R+

Γ1119905= Γ

1119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) = 1

forall119905 isin R+

(30)

where 1205750119895119905and 1205751

119895119905are Dirac measures defined by

1205750

119895119905= 1 minus 120575

1

119895119905=

1 if 119895 isin Γ0119905

0 if 119895 notin Γ0119905

forall119905 isin R+

1205751

119895119905= 1 minus 120575

0

119895119905=

1 if 119895 isin Γ1119905

0 if 119895 notin Γ1119905

forall119905 isin R+

(31)

Then one gets from (26) (29) and lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) =

lim120591rarr+infin

minus119867(120591) = 119867(+infinminus

) = 1 forall119905 isin R+ since 119896 isin (0 1)

that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+with 119909

119899+1isin [119879119909

119899]120572(119909119899)

forall119899 isin Z+since

lim119899rarrinfin

119865119909119899+1119909119899+2

(119905) ge ( lim119899rarrinfin

[

119899

prod

119894=1

119892 (119909119894 119909

119894+1 119896

minus119894+1

119905)])

sdot ( lim119899rarrinfin

11986511990911199092

(119896minus119899

119905)) forall119905 isin R+

(32)

Since lim119899rarrinfin

119865119909119899+1119909119899

(119905) = lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) = 1 forall119905 isin

R+and any given 119909

1isin 119883 then for any given 120576 isin R

+

and 120582 isin (0 1) there is 119873 = 119873(120576 120582) isin Z0+

such that119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+ Assume on the contrary

that there exist 119899119896

ge 119873 119899119896+2

gt 119899119896+1

gt 119899119896such that

119865119909119899119896+119894+1

119909119899119896+119894

(120576) ge 119865119909119899119896+119894+1

119909119899119896+119894

(1205762) gt 1 minus 120582 for 119894 = 0 1 and1minus120582 ge 119865

119909119899119896+2

119909119899119896

(120576)Then one has the following contradictionfor the subsequence 119909

119899119896

of 119909119899

1 minus 120582 ge 119865119909119899119896+2

119909119899119896

(120576)

ge Δ119872(119865

119909119899119896+2

119909119899119896+1

(

120576

2

) 119865119909119899119896+1

119909119899119896

(

120576

2

))

gt 1 minus 120582

(33)

Then 119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+

so that 119909119899 is

a Cauchy sequence Since (119883 F Δ119872) is complete one gets

119909119899 rarr 119909

lowast and 120572(119909119899) rarr 120572(119909

lowast

)

It is now proved that 119909lowast isin [119879119909lowast

]120572(119909lowast) Assume on the

contrary that 119909lowast notin [119879119909lowast

]120572(119909lowast) that is (119879119909lowast)(119909lowast) lt 120572(119909

lowast

)

and for some given 119910 isin [119879119909lowast

]120572(119909lowast) there is 119905

1= 119905

1(119910) isin R

+

such that 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) lt 1 for 119905 isin [0 1199051] Then since

119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+and since 119910 = 119909

lowast

1 gt 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119909lowast119910(119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

) 119865119909119899+1119910(

119905

4

)))

= Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

119905

4

) 119865[119879119909119899]120572(119909119899)

119910(

119905

4

)))

119905 isin [0 1199051]

(34)

If 119910 isin [119879119909lowast

]120572(119909lowast)is chosen to fulfill 119865

[119879119909119899]120572(119909119899)

119910(11990514) =

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(11990514) then

1 gt lim sup119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

1199051

2

) Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

119865[119879119909119899]120572(119909119899)

119910(

1199051

4

))) ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

1199051

2

)

Δ119872( lim119899rarrinfin

119865119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

))) = Δ119872(1

Δ119872(1 lim sup

119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)))

= lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)

= lim119899rarrinfin

119865[119879119909lowast]120572(119909lowast)[119879119909lowast]120572(119909lowast)

(

1199051

4

) = 1

for some 1199051isin R

+

(35)

a contradiction Then 119909lowast isin [119879119909lowast]120572(119909lowast)with 120572lowast = 120572(119909

lowast

) thatis it is a probabilistic 120572lowast-fuzzy fixed point of 119879 119883 rarr F(119883)

Corollary 3 Let F(119883) be the collection of all fuzzy sets ina PM-space (119883 F) where 119883 is a nonempty abstract set andlet 119879 119883 rarr F(119883) be a fuzzy mapping Assume that thecontractive condition (12) holds for some real constant 119896 isin

(0 1) subject to condition (2) of Theorem 2 and the specificparticular form of condition (3)

6 Journal of Function Spaces

(31015840

) there exists and strictly increasing sequence of nonnegativeintegers 119873

119899 which satisfies

119873119899+2minus1

prod

119895=119873119899+1

[

1198862(119909 119910 119896

minus119894+1

119905) + 1198863(119909 119910 119896

minus119894+1

119905)

1 minus 1198862(119909 119910 119896

minus119894+1119905)

] =

1

prod119873119899+1minus1

119895=119873119899

[(1198862(119909 119910 119896

minus119894+1119905) + 119886

3(119909 119910 119896

minus119894+1119905)) (1 minus 119886

2(119909 119910 119896

minus119894+1119905))]

forall119905 isin R+ forall119909 isin 119883 119910 isin [119879119909]

120572(119909)sub 119862119861 (119883) forall119899 isin Z

0+

(36)

Proof It is direct from that of Theorem 2 since condition(31015840) guarantees condition (3) of Theorem 2 for any finite firstelement 119873

0isin Z

0+of a strictly increasing sequence 119873

119899

subject to 0 lt 1198721le 119873

119899+1minus 119873

119899le 119872

2lt +infin

Corollary 4 Theorem 2 and Corollary 3 also hold if thecontractive condition (1) is modified as follows

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905)

ge 1198861(119909

1 119909

2 119896119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) + 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905)

= (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

forall119905 isin R+ forall119909

1= 119909 isin 119883 119909

119899+1isin [119879119909

119899]120572(119909119899)sub 119862119861 (119883) forall119899 isin Z

0+

(37)

Proof It is direct from that of Theorem 2 since the abovecontraction condition follows from (12) as a lower-boundwhich has been used in the proof of Theorem 2

Corollary 5 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of119883 and

each sequence 119909119899 sub 119883 of the form 119909

1isin 119883 119909

119899+1isin

[119879119909119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+which satisfies for some

real constant 119896 isin (0 1) the contractive constraint

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge min (1 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905) + 1198863(119909 119910 119896119905) 119865

119909119910(119905))

(38)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894 119883 times 119883 times R

0+rarr [0 1] for

119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Then a sequence 119909119899 may be built for any given arbitrary

1199091= 119909 isin 119883 satisfying 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] and lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isin R+

If in addition (119883 F) is endowed with the minimumtriangular norm Δ

119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each above sequence 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Define the indicator function sequence 120590119894 119883 times 119883 times

R0+

rarr 0 1 119894 isin Z+as

120590119894(119909

119894 119909

119894+1 119896

minus119894+1

119905)

=

1 if 119892 (119909119894 119909

119894+1 119896

minus119894+1

119905) 119865119909119894119909119894+1

(119896minus119894

119905) le 1

0 otherwise

forall119905 isin R+

(39)

with 119892(119909119894 119909

119894+1 119905) defined as in Theorem 2 Then even if for

some 119895 isin Z+and all 119894(ge 119895) isin Z

+ 120590

119894(119909

119894 119909

119894+1 119905) = 0 because

119892(119909119894 119909

119894+1 119896

minus119894+1

119905)119865119909119894119909119894+1

(119896minus119894

119905) gt 1 forall119894(ge 119895) isin Z+ it follows

from (38)-(39) and (26) that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

with 119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+ The rest of the proof is close

to that of Theorem 2

Example 6 It is claimed through this example to revisitthe idea of fuzzy fixed point addressed in [4ndash6 32] tothat of probabilistic fuzzy fixed point according to thedefinition and the formalism given above Consider 119883 =

06 07 08 06 07 08 and let 119879 119883 rarr F(119883) be aprobabilistic fuzzy mapping defined as follows

119879 (06) (119905) = 119879 (08) (119905) =

3

4

if 119905 = 08

1

2

if 119905 = 07

0 if 119905 = 06

119879 (07) (119905) =

0 if 119905 = 08

1

3

if 119905 = 07

3

4

if 119905 = 06

(40)

The 120572-level sets are

[11987906]12

= [11987908]12

= 07 08

[11987906]34

= [11987908]34

= 08

[11987907]34

= 06

[11987907]13

= 07

(41)

Journal of Function Spaces 7

Note that 119865119909[119879119909]

120572(119909)

(119905) = sup(119865119909119910(119905) 119910 isin [119879119909]

120572(119909)) for any

119909 isin 119883 and 119905 isin R and some 120572(119909) isin (0 1] The probabilitydensity functions are as follows with 119865

119909119910(119905) = 119905(119905 + |119909 minus 119910|)

forall119909 119910 isin 119883 forall119905 isin R+

11986506[11987906]

34

(119905) = 1198650608

(119905) =

119905

119905 + 02

forall119905 isin R+

11986506[11987906]

12

(119905) = 1198650607

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

12

(119905) = 1198650807

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

34

(119905) = 1198650808

(119905) = 1 forall119905 isin R+

11986507[11987907]

34

(119905) = 1198650706

(119905) =

119905

119905 + 01

forall119905 isin R+

11986507[11987907]

13

(119905) = 1198650707

(119905) = 1 forall119905 isin R+

119867[119879119909]12[119879119909]12

(119905) =

119905

119905 + 01

for 119909 = 06 08 forall119905 isin R+

119867[119879119909]34[119879119909]34

(119905) = 1 for 119909 = 06 08 forall119905 isin R+

119867[119879119909]12[119879119910]34

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

13

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

34

(119905) =

119905

119905 + 02

for 119909 119910 = 06 08 forall119905 isin R+

119867[11987907]

13[11987907]

13

(119905) = 1 forall119905 isin R+

119867[11987907]

34[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

119867[11987907]

13[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

(42)

Assume that the contractive condition of Theorem 2 holdsunder the form

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905) ge 120572 (119909119899 119909

119899+1 119905 119899)

sdot [119865119909119899[119879119909119899]120572(119909119899)

(119896minus1

119905) + 119865119909119899+1[119879119909119899+1]120572(119909119899+1)

(119896minus1

119905)

+ 119865119909119899119909119899+1

(119896minus1

119905)] 119899 isin Z0+

(43)

for sequences 119909119899 sub 119883 with initial points 119909

0= 119909 isin 119883 119909

1isin

[1198791199090]120572 where 120572(119909 119910 119905 119899) = 120572

119894(119909 119910 119905 119899) 119894 = 1 2 3 forall119905 isin R

+

satisfies

120572 (119909 119910 119905 119899) = 120572119886(119905) le

119896119905

3 (119896119905 + 02)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

120572 (119909 119910 119905 119899) = 120572119887(119905) le

1

3

(1 minus 119890minus120582119905

)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

(44)

for some ℓ(ge 2) isin Z0+ some 119896 isin (0 1) and some 120582 isin R

+

Note that [11987907]13

= 07 with 11986507[11987907]

13

(119905) = 1198650707

(119905) =

1 forall119905 isin R+is a probabilistic 13-fuzzy fixed point of 119879

119883 rarr F(119883) to which the sequences 06 08 07 07 08 06 07 07 converge

3 Further Results

Note that the uniqueness of probabilistic fuzzy fixed points isnot an interesting property to study in the context of prob-abilistic fuzzy fixed point since distinct level sets associatedwith mappings of the form 119879 119883 rarr F(119883) can easily haveintersections of cardinal greater than one in many problemsin the fuzzy context The subsequent result gives conditionsfor the case when several distinct probabilistic fuzzy fixedpoints if they exist are in the intersections of their respectivelevel sets under slightly extended contractive conditions ofthose given in Section 2The extended conditions are of directapplicability

Theorem 7 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 with an

extended contractive condition (12) such that it also holds forany 119909 119910 isin 119883 being probabilistic 120572(119909lowast)-fuzzy fixed points119909 = 119909

lowast

isin [119879119909lowast

]120572(119909lowast)and 119910 = 119910

lowast

isin [119879119910lowast

]120572(119910lowast)of 119879 119883 rarr

F(119883) Then the set [119879119910lowast]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)is nonempty and

119909lowast

119910lowast

isin ([119879119910lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast))

Proof If119909lowast = 119910lowast the result is obvious Assume that there exista probabilistic 120572(119909lowast)-fuzzy fixed point 119909lowast isin [119879119909lowast]

120572(119909lowast)and a

probabilistic 120572(119910lowast)-fuzzy fixed point 119910lowast( = 119909lowast

) isin [119879119910lowast

]120572(119910lowast)

Consider two convergent sequences 119909119899 rarr 119909

lowast and 119910119899 rarr

119910lowast in119883 Then

119865119909119899119910119899

(119905) ge Δ119872(119865

119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

119865119910119899119910lowast (

119905

4

))) forall119905 isin R+

lim inf119899rarrinfin

119865119909119899119910119899

(119905) ge lim inf119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909lowast119910lowast (

119905

4

) 119865119910119899119910lowast (

119905

4

)))

ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

lim119899rarrinfin

119865119910119899119910lowast (

119905

4

))) = Δ119872(1 Δ

119872(119865

119909lowast119910lowast (

119905

4

) 1))

ge 119865119909lowast119910lowast (

119905

4

) forall119905 isin R+

(45)

Assume that 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+

Then from the extended contractive condition (12) and since1 = 119865

119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905) ge 119865119909lowast119910lowast(119905) forall119905 isin R

+

one gets for some 119911119909isin [119879119909

lowast

]120572(119909lowast)and 119911

119910isin [119879119910

lowast

]120572(119910lowast)

8 Journal of Function Spaces

since [119879119909lowast]120572(119909lowast)and [119879119910lowast]

120572(119910lowast)are members of 119862119861(119883) that

is nonempty closed and bounded sets

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905)

= min (119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905))

= min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge 1198861(119909

lowast

119910lowast

119896119905)

sdot 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) + 1198862(119909

lowast

119910lowast

119896119905) 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905)

+ 1198863(119909

lowast

119910lowast

119896119905) 119865119909lowast119910lowast (119905) ge (119886

1(119909

lowast

119910lowast

119896119905)

+ 1198862(119909

lowast

119910lowast

119896119905) + 1198863(119909

lowast

119910lowast

119896119905)) 119865119909lowast119910lowast (119905)

(46)

so that

min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) 119865119909lowast119910lowast (119896

minus1

119905)

ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) Δ119872(119865

119909lowast119911119910

(

119896minus1

119905

2

) 119865119911119910119910lowast (

119896minus1

119905

2

))

ge

119898

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)] Δ119872(119865

119909lowast119911119910

((2119896)minus119898

119905) 119865119911119910119910lowast ((2119896)

119898

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot lim inf119899rarrinfin

Δ119872(119865

119909lowast119911119910

((2119896)minus119899

119905) 119865119911119910119910lowast ((2119896)

minus119899

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot Δ119872(119865

119909lowast119911119910

( lim119899rarrinfin

(2119896)minus119899

119905) 119865119911119910119910lowast ( lim

119899rarrinfin

(2119896)minus119899

119905))

= lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])119867 (+infinminus

) = 1 forall119905 isin R+ forall119898 isin Z

+

(47)

and then 119910lowast = 119911119909 119909lowast = 119911

119910([119879119910

lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)) so that

[119879119910lowast

]120572(119910lowast)cap[119879119909

lowast

]120572(119909lowast)is nonemptyWe can now assume that

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) le 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+ It is direct to

prove in a close way to the above proof that 119910lowast isin [119879119910lowast]120572(119910lowast)cap

[119879119909lowast

]120572(119909lowast)

Remark 8 Note that a direct consequence of Theorem 7from the definition of the level sets is that

119909lowast

119910lowast

isin (( ⋂

120573isin(0120572(119909lowast)]

[119879119909lowast

]120573) cap ( ⋂

120573isin(0120572(119910lowast)]

[119879119910lowast

]120573))

(48)

if 119909lowast isin [119879119909lowast

]120572(119909lowast)and 119910lowast isin [119879119910

lowast

]120572(119910lowast)are any probabilistic

120572(119909lowast

) and 120572(119910lowast)-fuzzy fixed points

Remark 9 Note that corollaries to Theorem 7 can also bestated ldquomutatis-mutandisrdquo under extendedcontractiveconditions

for probabilistic fuzzy fixed points to those given in Corollar-ies 3ndash5

Theorem 10 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 and condi-

tions (1)ndash(3) where the contractive condition

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(49)

is extended to hold for any 119909 119910 isin 119883 with 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Proof Note from Theorem 2 a sequence 119909119899 sub 119883 can be

built being a (convergent) Cauchy sequence such that 119909119899 rarr

Journal of Function Spaces 9

119909lowast 119909

119899+1= 119879119909

119899isin [119879119909

119899]120572(119909119899) forall119899 isin Z

0+for any given 119909

0isin 119883

Then

lim119899rarrinfin

119865119909119899119879119909119899

(119905) = lim119899rarrinfin

119865119909119899119909lowast (119905) = 1 forall119905 isin R

+

lim119899rarrinfin

119865119879119909119899119909lowast (119905) = 1 forall119905 isin R

+

since 119865119879119909119899119909lowast (119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

)) 119865119879119909119899119879119909119899

(

119905

4

))

gt min (1 minus 120582 1 minus 120582 1) = 1 minus 120582

(50)

for any given 119905 isin R+ 120582 isin (0 1) 119899(ge 119873

0) isin Z

0+ and 119873

0=

1198730(120576 120582) ge max(119873

1 119873

2) such that

119865119909lowast119909119899

(

119905

2

) gt 1 minus 120582

for 119899 (ge 1198731) isin Z

0+and some 119873

1= 119873

1(120576 120582) isin Z

0+

119865119909119899+1119909119899

(

119905

4

) gt 1 minus 120582

for 119899 (ge 1198732) isin Z

0+and some 119873

2= 119873

2(120576 120582) isin Z

0+

(51)

since 119865119879119909119899119879119909119899

(1199054) = 1 forall119905 isin R+ forall119899 isin Z

0+from property (1)

of (3) for PM-spaces

Theorem 11 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

(52)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894isin [0 1] for 119894 =

1 3 4 5 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 =

5

sum

119894=1

119886119894le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(53)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(54)

Then a sequence 119909119899 may be constructed for any given

arbitrary 1199091= 119909 isin 119883 such that 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883)

with 120572(119909119899) sube (0 1] satisfying lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isinR+If in addition (119883 F) is endowed with the minimum

triangular norm Δ119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Since [119879119909]120572(119909)

[119879119910]120572(119910)

sub 119883

max (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min119911isin[119879119909]

120572(119909)120596isin[119879119910]

120572(119910)

(119865119911[119879119910]

120572(119910)

(119896119905) 119865120596[119879119909]

120572(119909)

(119896119905))

= 119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905) + 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

forall119905 isin R+ forall119909 119910 isin 119883

(55)

Now for any given 119909 119910 isin 119883 Then the following cases canoccur

(a) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905) forall119905 isin R+

(1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4) 119865

119910[119879119910]120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4+ 119886

5) 119865

119909119910(119905) forall119905 isin R

+

(56)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(57)

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Journal of Function Spaces 5

then 120575(Γ1119905)120575(Γ

0119905) = +infin forall119905 isin R

+ forall119894 isin Z

+ where 120575(Γ

0119905) =

prodinfin

119895=0[1205750

119895119905] = prod

infin

119895=0[1 minus 120575

1

119895119905] and 120575(Γ

1119905) = prod

infin

119895=0[1205751

119895119905] = prod

infin

119895=0[1 minus

1205750

119895119905] forall119905 isin R

+are discrete measures of the subsequent sets

Γ0119905= Γ

0119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) isin [0 1)

forall119905 isin R+

Γ1119905= Γ

1119905(119909

119895)

= 119895 = 119895 (119905) isin Z0+ 119892 (119909

119895 119909

119895+1 119905) = 1

forall119905 isin R+

(30)

where 1205750119895119905and 1205751

119895119905are Dirac measures defined by

1205750

119895119905= 1 minus 120575

1

119895119905=

1 if 119895 isin Γ0119905

0 if 119895 notin Γ0119905

forall119905 isin R+

1205751

119895119905= 1 minus 120575

0

119895119905=

1 if 119895 isin Γ1119905

0 if 119895 notin Γ1119905

forall119905 isin R+

(31)

Then one gets from (26) (29) and lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) =

lim120591rarr+infin

minus119867(120591) = 119867(+infinminus

) = 1 forall119905 isin R+ since 119896 isin (0 1)

that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+with 119909

119899+1isin [119879119909

119899]120572(119909119899)

forall119899 isin Z+since

lim119899rarrinfin

119865119909119899+1119909119899+2

(119905) ge ( lim119899rarrinfin

[

119899

prod

119894=1

119892 (119909119894 119909

119894+1 119896

minus119894+1

119905)])

sdot ( lim119899rarrinfin

11986511990911199092

(119896minus119899

119905)) forall119905 isin R+

(32)

Since lim119899rarrinfin

119865119909119899+1119909119899

(119905) = lim119899rarrinfin

11986511990911199092

(119896minus119899

119905) = 1 forall119905 isin

R+and any given 119909

1isin 119883 then for any given 120576 isin R

+

and 120582 isin (0 1) there is 119873 = 119873(120576 120582) isin Z0+

such that119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+ Assume on the contrary

that there exist 119899119896

ge 119873 119899119896+2

gt 119899119896+1

gt 119899119896such that

119865119909119899119896+119894+1

119909119899119896+119894

(120576) ge 119865119909119899119896+119894+1

119909119899119896+119894

(1205762) gt 1 minus 120582 for 119894 = 0 1 and1minus120582 ge 119865

119909119899119896+2

119909119899119896

(120576)Then one has the following contradictionfor the subsequence 119909

119899119896

of 119909119899

1 minus 120582 ge 119865119909119899119896+2

119909119899119896

(120576)

ge Δ119872(119865

119909119899119896+2

119909119899119896+1

(

120576

2

) 119865119909119899119896+1

119909119899119896

(

120576

2

))

gt 1 minus 120582

(33)

Then 119865119909119899+1119909119899

(120576) gt 1 minus 120582 forall119899(ge 119873) isin Z0+

so that 119909119899 is

a Cauchy sequence Since (119883 F Δ119872) is complete one gets

119909119899 rarr 119909

lowast and 120572(119909119899) rarr 120572(119909

lowast

)

It is now proved that 119909lowast isin [119879119909lowast

]120572(119909lowast) Assume on the

contrary that 119909lowast notin [119879119909lowast

]120572(119909lowast) that is (119879119909lowast)(119909lowast) lt 120572(119909

lowast

)

and for some given 119910 isin [119879119909lowast

]120572(119909lowast) there is 119905

1= 119905

1(119910) isin R

+

such that 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) lt 1 for 119905 isin [0 1199051] Then since

119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+and since 119910 = 119909

lowast

1 gt 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119909lowast119910(119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

) 119865119909119899+1119910(

119905

4

)))

= Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

119905

4

) 119865[119879119909119899]120572(119909119899)

119910(

119905

4

)))

119905 isin [0 1199051]

(34)

If 119910 isin [119879119909lowast

]120572(119909lowast)is chosen to fulfill 119865

[119879119909119899]120572(119909119899)

119910(11990514) =

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(11990514) then

1 gt lim sup119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

1199051

2

) Δ119872(119865

119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

119865[119879119909119899]120572(119909119899)

119910(

1199051

4

))) ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

1199051

2

)

Δ119872( lim119899rarrinfin

119865119909119899[119879119909119899]120572(119909119899)

(

1199051

4

)

lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

))) = Δ119872(1

Δ119872(1 lim sup

119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)))

= lim sup119899rarrinfin

119865[119879119909119899]120572(119909119899)

[119879119909lowast]120572(119909lowast)

(

1199051

4

)

= lim119899rarrinfin

119865[119879119909lowast]120572(119909lowast)[119879119909lowast]120572(119909lowast)

(

1199051

4

) = 1

for some 1199051isin R

+

(35)

a contradiction Then 119909lowast isin [119879119909lowast]120572(119909lowast)with 120572lowast = 120572(119909

lowast

) thatis it is a probabilistic 120572lowast-fuzzy fixed point of 119879 119883 rarr F(119883)

Corollary 3 Let F(119883) be the collection of all fuzzy sets ina PM-space (119883 F) where 119883 is a nonempty abstract set andlet 119879 119883 rarr F(119883) be a fuzzy mapping Assume that thecontractive condition (12) holds for some real constant 119896 isin

(0 1) subject to condition (2) of Theorem 2 and the specificparticular form of condition (3)

6 Journal of Function Spaces

(31015840

) there exists and strictly increasing sequence of nonnegativeintegers 119873

119899 which satisfies

119873119899+2minus1

prod

119895=119873119899+1

[

1198862(119909 119910 119896

minus119894+1

119905) + 1198863(119909 119910 119896

minus119894+1

119905)

1 minus 1198862(119909 119910 119896

minus119894+1119905)

] =

1

prod119873119899+1minus1

119895=119873119899

[(1198862(119909 119910 119896

minus119894+1119905) + 119886

3(119909 119910 119896

minus119894+1119905)) (1 minus 119886

2(119909 119910 119896

minus119894+1119905))]

forall119905 isin R+ forall119909 isin 119883 119910 isin [119879119909]

120572(119909)sub 119862119861 (119883) forall119899 isin Z

0+

(36)

Proof It is direct from that of Theorem 2 since condition(31015840) guarantees condition (3) of Theorem 2 for any finite firstelement 119873

0isin Z

0+of a strictly increasing sequence 119873

119899

subject to 0 lt 1198721le 119873

119899+1minus 119873

119899le 119872

2lt +infin

Corollary 4 Theorem 2 and Corollary 3 also hold if thecontractive condition (1) is modified as follows

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905)

ge 1198861(119909

1 119909

2 119896119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) + 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905)

= (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

forall119905 isin R+ forall119909

1= 119909 isin 119883 119909

119899+1isin [119879119909

119899]120572(119909119899)sub 119862119861 (119883) forall119899 isin Z

0+

(37)

Proof It is direct from that of Theorem 2 since the abovecontraction condition follows from (12) as a lower-boundwhich has been used in the proof of Theorem 2

Corollary 5 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of119883 and

each sequence 119909119899 sub 119883 of the form 119909

1isin 119883 119909

119899+1isin

[119879119909119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+which satisfies for some

real constant 119896 isin (0 1) the contractive constraint

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge min (1 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905) + 1198863(119909 119910 119896119905) 119865

119909119910(119905))

(38)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894 119883 times 119883 times R

0+rarr [0 1] for

119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Then a sequence 119909119899 may be built for any given arbitrary

1199091= 119909 isin 119883 satisfying 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] and lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isin R+

If in addition (119883 F) is endowed with the minimumtriangular norm Δ

119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each above sequence 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Define the indicator function sequence 120590119894 119883 times 119883 times

R0+

rarr 0 1 119894 isin Z+as

120590119894(119909

119894 119909

119894+1 119896

minus119894+1

119905)

=

1 if 119892 (119909119894 119909

119894+1 119896

minus119894+1

119905) 119865119909119894119909119894+1

(119896minus119894

119905) le 1

0 otherwise

forall119905 isin R+

(39)

with 119892(119909119894 119909

119894+1 119905) defined as in Theorem 2 Then even if for

some 119895 isin Z+and all 119894(ge 119895) isin Z

+ 120590

119894(119909

119894 119909

119894+1 119905) = 0 because

119892(119909119894 119909

119894+1 119896

minus119894+1

119905)119865119909119894119909119894+1

(119896minus119894

119905) gt 1 forall119894(ge 119895) isin Z+ it follows

from (38)-(39) and (26) that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

with 119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+ The rest of the proof is close

to that of Theorem 2

Example 6 It is claimed through this example to revisitthe idea of fuzzy fixed point addressed in [4ndash6 32] tothat of probabilistic fuzzy fixed point according to thedefinition and the formalism given above Consider 119883 =

06 07 08 06 07 08 and let 119879 119883 rarr F(119883) be aprobabilistic fuzzy mapping defined as follows

119879 (06) (119905) = 119879 (08) (119905) =

3

4

if 119905 = 08

1

2

if 119905 = 07

0 if 119905 = 06

119879 (07) (119905) =

0 if 119905 = 08

1

3

if 119905 = 07

3

4

if 119905 = 06

(40)

The 120572-level sets are

[11987906]12

= [11987908]12

= 07 08

[11987906]34

= [11987908]34

= 08

[11987907]34

= 06

[11987907]13

= 07

(41)

Journal of Function Spaces 7

Note that 119865119909[119879119909]

120572(119909)

(119905) = sup(119865119909119910(119905) 119910 isin [119879119909]

120572(119909)) for any

119909 isin 119883 and 119905 isin R and some 120572(119909) isin (0 1] The probabilitydensity functions are as follows with 119865

119909119910(119905) = 119905(119905 + |119909 minus 119910|)

forall119909 119910 isin 119883 forall119905 isin R+

11986506[11987906]

34

(119905) = 1198650608

(119905) =

119905

119905 + 02

forall119905 isin R+

11986506[11987906]

12

(119905) = 1198650607

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

12

(119905) = 1198650807

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

34

(119905) = 1198650808

(119905) = 1 forall119905 isin R+

11986507[11987907]

34

(119905) = 1198650706

(119905) =

119905

119905 + 01

forall119905 isin R+

11986507[11987907]

13

(119905) = 1198650707

(119905) = 1 forall119905 isin R+

119867[119879119909]12[119879119909]12

(119905) =

119905

119905 + 01

for 119909 = 06 08 forall119905 isin R+

119867[119879119909]34[119879119909]34

(119905) = 1 for 119909 = 06 08 forall119905 isin R+

119867[119879119909]12[119879119910]34

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

13

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

34

(119905) =

119905

119905 + 02

for 119909 119910 = 06 08 forall119905 isin R+

119867[11987907]

13[11987907]

13

(119905) = 1 forall119905 isin R+

119867[11987907]

34[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

119867[11987907]

13[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

(42)

Assume that the contractive condition of Theorem 2 holdsunder the form

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905) ge 120572 (119909119899 119909

119899+1 119905 119899)

sdot [119865119909119899[119879119909119899]120572(119909119899)

(119896minus1

119905) + 119865119909119899+1[119879119909119899+1]120572(119909119899+1)

(119896minus1

119905)

+ 119865119909119899119909119899+1

(119896minus1

119905)] 119899 isin Z0+

(43)

for sequences 119909119899 sub 119883 with initial points 119909

0= 119909 isin 119883 119909

1isin

[1198791199090]120572 where 120572(119909 119910 119905 119899) = 120572

119894(119909 119910 119905 119899) 119894 = 1 2 3 forall119905 isin R

+

satisfies

120572 (119909 119910 119905 119899) = 120572119886(119905) le

119896119905

3 (119896119905 + 02)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

120572 (119909 119910 119905 119899) = 120572119887(119905) le

1

3

(1 minus 119890minus120582119905

)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

(44)

for some ℓ(ge 2) isin Z0+ some 119896 isin (0 1) and some 120582 isin R

+

Note that [11987907]13

= 07 with 11986507[11987907]

13

(119905) = 1198650707

(119905) =

1 forall119905 isin R+is a probabilistic 13-fuzzy fixed point of 119879

119883 rarr F(119883) to which the sequences 06 08 07 07 08 06 07 07 converge

3 Further Results

Note that the uniqueness of probabilistic fuzzy fixed points isnot an interesting property to study in the context of prob-abilistic fuzzy fixed point since distinct level sets associatedwith mappings of the form 119879 119883 rarr F(119883) can easily haveintersections of cardinal greater than one in many problemsin the fuzzy context The subsequent result gives conditionsfor the case when several distinct probabilistic fuzzy fixedpoints if they exist are in the intersections of their respectivelevel sets under slightly extended contractive conditions ofthose given in Section 2The extended conditions are of directapplicability

Theorem 7 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 with an

extended contractive condition (12) such that it also holds forany 119909 119910 isin 119883 being probabilistic 120572(119909lowast)-fuzzy fixed points119909 = 119909

lowast

isin [119879119909lowast

]120572(119909lowast)and 119910 = 119910

lowast

isin [119879119910lowast

]120572(119910lowast)of 119879 119883 rarr

F(119883) Then the set [119879119910lowast]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)is nonempty and

119909lowast

119910lowast

isin ([119879119910lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast))

Proof If119909lowast = 119910lowast the result is obvious Assume that there exista probabilistic 120572(119909lowast)-fuzzy fixed point 119909lowast isin [119879119909lowast]

120572(119909lowast)and a

probabilistic 120572(119910lowast)-fuzzy fixed point 119910lowast( = 119909lowast

) isin [119879119910lowast

]120572(119910lowast)

Consider two convergent sequences 119909119899 rarr 119909

lowast and 119910119899 rarr

119910lowast in119883 Then

119865119909119899119910119899

(119905) ge Δ119872(119865

119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

119865119910119899119910lowast (

119905

4

))) forall119905 isin R+

lim inf119899rarrinfin

119865119909119899119910119899

(119905) ge lim inf119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909lowast119910lowast (

119905

4

) 119865119910119899119910lowast (

119905

4

)))

ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

lim119899rarrinfin

119865119910119899119910lowast (

119905

4

))) = Δ119872(1 Δ

119872(119865

119909lowast119910lowast (

119905

4

) 1))

ge 119865119909lowast119910lowast (

119905

4

) forall119905 isin R+

(45)

Assume that 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+

Then from the extended contractive condition (12) and since1 = 119865

119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905) ge 119865119909lowast119910lowast(119905) forall119905 isin R

+

one gets for some 119911119909isin [119879119909

lowast

]120572(119909lowast)and 119911

119910isin [119879119910

lowast

]120572(119910lowast)

8 Journal of Function Spaces

since [119879119909lowast]120572(119909lowast)and [119879119910lowast]

120572(119910lowast)are members of 119862119861(119883) that

is nonempty closed and bounded sets

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905)

= min (119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905))

= min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge 1198861(119909

lowast

119910lowast

119896119905)

sdot 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) + 1198862(119909

lowast

119910lowast

119896119905) 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905)

+ 1198863(119909

lowast

119910lowast

119896119905) 119865119909lowast119910lowast (119905) ge (119886

1(119909

lowast

119910lowast

119896119905)

+ 1198862(119909

lowast

119910lowast

119896119905) + 1198863(119909

lowast

119910lowast

119896119905)) 119865119909lowast119910lowast (119905)

(46)

so that

min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) 119865119909lowast119910lowast (119896

minus1

119905)

ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) Δ119872(119865

119909lowast119911119910

(

119896minus1

119905

2

) 119865119911119910119910lowast (

119896minus1

119905

2

))

ge

119898

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)] Δ119872(119865

119909lowast119911119910

((2119896)minus119898

119905) 119865119911119910119910lowast ((2119896)

119898

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot lim inf119899rarrinfin

Δ119872(119865

119909lowast119911119910

((2119896)minus119899

119905) 119865119911119910119910lowast ((2119896)

minus119899

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot Δ119872(119865

119909lowast119911119910

( lim119899rarrinfin

(2119896)minus119899

119905) 119865119911119910119910lowast ( lim

119899rarrinfin

(2119896)minus119899

119905))

= lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])119867 (+infinminus

) = 1 forall119905 isin R+ forall119898 isin Z

+

(47)

and then 119910lowast = 119911119909 119909lowast = 119911

119910([119879119910

lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)) so that

[119879119910lowast

]120572(119910lowast)cap[119879119909

lowast

]120572(119909lowast)is nonemptyWe can now assume that

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) le 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+ It is direct to

prove in a close way to the above proof that 119910lowast isin [119879119910lowast]120572(119910lowast)cap

[119879119909lowast

]120572(119909lowast)

Remark 8 Note that a direct consequence of Theorem 7from the definition of the level sets is that

119909lowast

119910lowast

isin (( ⋂

120573isin(0120572(119909lowast)]

[119879119909lowast

]120573) cap ( ⋂

120573isin(0120572(119910lowast)]

[119879119910lowast

]120573))

(48)

if 119909lowast isin [119879119909lowast

]120572(119909lowast)and 119910lowast isin [119879119910

lowast

]120572(119910lowast)are any probabilistic

120572(119909lowast

) and 120572(119910lowast)-fuzzy fixed points

Remark 9 Note that corollaries to Theorem 7 can also bestated ldquomutatis-mutandisrdquo under extendedcontractiveconditions

for probabilistic fuzzy fixed points to those given in Corollar-ies 3ndash5

Theorem 10 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 and condi-

tions (1)ndash(3) where the contractive condition

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(49)

is extended to hold for any 119909 119910 isin 119883 with 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Proof Note from Theorem 2 a sequence 119909119899 sub 119883 can be

built being a (convergent) Cauchy sequence such that 119909119899 rarr

Journal of Function Spaces 9

119909lowast 119909

119899+1= 119879119909

119899isin [119879119909

119899]120572(119909119899) forall119899 isin Z

0+for any given 119909

0isin 119883

Then

lim119899rarrinfin

119865119909119899119879119909119899

(119905) = lim119899rarrinfin

119865119909119899119909lowast (119905) = 1 forall119905 isin R

+

lim119899rarrinfin

119865119879119909119899119909lowast (119905) = 1 forall119905 isin R

+

since 119865119879119909119899119909lowast (119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

)) 119865119879119909119899119879119909119899

(

119905

4

))

gt min (1 minus 120582 1 minus 120582 1) = 1 minus 120582

(50)

for any given 119905 isin R+ 120582 isin (0 1) 119899(ge 119873

0) isin Z

0+ and 119873

0=

1198730(120576 120582) ge max(119873

1 119873

2) such that

119865119909lowast119909119899

(

119905

2

) gt 1 minus 120582

for 119899 (ge 1198731) isin Z

0+and some 119873

1= 119873

1(120576 120582) isin Z

0+

119865119909119899+1119909119899

(

119905

4

) gt 1 minus 120582

for 119899 (ge 1198732) isin Z

0+and some 119873

2= 119873

2(120576 120582) isin Z

0+

(51)

since 119865119879119909119899119879119909119899

(1199054) = 1 forall119905 isin R+ forall119899 isin Z

0+from property (1)

of (3) for PM-spaces

Theorem 11 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

(52)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894isin [0 1] for 119894 =

1 3 4 5 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 =

5

sum

119894=1

119886119894le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(53)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(54)

Then a sequence 119909119899 may be constructed for any given

arbitrary 1199091= 119909 isin 119883 such that 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883)

with 120572(119909119899) sube (0 1] satisfying lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isinR+If in addition (119883 F) is endowed with the minimum

triangular norm Δ119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Since [119879119909]120572(119909)

[119879119910]120572(119910)

sub 119883

max (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min119911isin[119879119909]

120572(119909)120596isin[119879119910]

120572(119910)

(119865119911[119879119910]

120572(119910)

(119896119905) 119865120596[119879119909]

120572(119909)

(119896119905))

= 119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905) + 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

forall119905 isin R+ forall119909 119910 isin 119883

(55)

Now for any given 119909 119910 isin 119883 Then the following cases canoccur

(a) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905) forall119905 isin R+

(1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4) 119865

119910[119879119910]120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4+ 119886

5) 119865

119909119910(119905) forall119905 isin R

+

(56)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(57)

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces

(31015840

) there exists and strictly increasing sequence of nonnegativeintegers 119873

119899 which satisfies

119873119899+2minus1

prod

119895=119873119899+1

[

1198862(119909 119910 119896

minus119894+1

119905) + 1198863(119909 119910 119896

minus119894+1

119905)

1 minus 1198862(119909 119910 119896

minus119894+1119905)

] =

1

prod119873119899+1minus1

119895=119873119899

[(1198862(119909 119910 119896

minus119894+1119905) + 119886

3(119909 119910 119896

minus119894+1119905)) (1 minus 119886

2(119909 119910 119896

minus119894+1119905))]

forall119905 isin R+ forall119909 isin 119883 119910 isin [119879119909]

120572(119909)sub 119862119861 (119883) forall119899 isin Z

0+

(36)

Proof It is direct from that of Theorem 2 since condition(31015840) guarantees condition (3) of Theorem 2 for any finite firstelement 119873

0isin Z

0+of a strictly increasing sequence 119873

119899

subject to 0 lt 1198721le 119873

119899+1minus 119873

119899le 119872

2lt +infin

Corollary 4 Theorem 2 and Corollary 3 also hold if thecontractive condition (1) is modified as follows

119867[1198791199091]120572(1199091)[1198791199092]120572(1199092)

(119896119905)

ge 1198861(119909

1 119909

2 119896119905) 119865

1199091[1198791199091]120572(1199091)

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905) + 1198863(119909

1 119909

2 119896119905) 119865

11990911199092

(119905)

= (1198861(119909

1 119909

2 119896119905) + 119886

3(119909

1 119909

2 119896119905)) 119865

11990911199092

(119905)

+ 1198862(119909

1 119909

2 119896119905) 119865

1199092[1198791199092]120572(1199092)

(119896119905)

forall119905 isin R+ forall119909

1= 119909 isin 119883 119909

119899+1isin [119879119909

119899]120572(119909119899)sub 119862119861 (119883) forall119899 isin Z

0+

(37)

Proof It is direct from that of Theorem 2 since the abovecontraction condition follows from (12) as a lower-boundwhich has been used in the proof of Theorem 2

Corollary 5 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of119883 and

each sequence 119909119899 sub 119883 of the form 119909

1isin 119883 119909

119899+1isin

[119879119909119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+which satisfies for some

real constant 119896 isin (0 1) the contractive constraint

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge min (1 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905) + 1198863(119909 119910 119896119905) 119865

119909119910(119905))

(38)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894 119883 times 119883 times R

0+rarr [0 1] for

119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Then a sequence 119909119899 may be built for any given arbitrary

1199091= 119909 isin 119883 satisfying 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883) with

120572(119909119899) sube (0 1] and lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isin R+

If in addition (119883 F) is endowed with the minimumtriangular norm Δ

119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each above sequence 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Define the indicator function sequence 120590119894 119883 times 119883 times

R0+

rarr 0 1 119894 isin Z+as

120590119894(119909

119894 119909

119894+1 119896

minus119894+1

119905)

=

1 if 119892 (119909119894 119909

119894+1 119896

minus119894+1

119905) 119865119909119894119909119894+1

(119896minus119894

119905) le 1

0 otherwise

forall119905 isin R+

(39)

with 119892(119909119894 119909

119894+1 119905) defined as in Theorem 2 Then even if for

some 119895 isin Z+and all 119894(ge 119895) isin Z

+ 120590

119894(119909

119894 119909

119894+1 119905) = 0 because

119892(119909119894 119909

119894+1 119896

minus119894+1

119905)119865119909119894119909119894+1

(119896minus119894

119905) gt 1 forall119894(ge 119895) isin Z+ it follows

from (38)-(39) and (26) that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

with 119909119899+1

isin [119879119909119899]120572(119909119899) forall119899 isin Z

+ The rest of the proof is close

to that of Theorem 2

Example 6 It is claimed through this example to revisitthe idea of fuzzy fixed point addressed in [4ndash6 32] tothat of probabilistic fuzzy fixed point according to thedefinition and the formalism given above Consider 119883 =

06 07 08 06 07 08 and let 119879 119883 rarr F(119883) be aprobabilistic fuzzy mapping defined as follows

119879 (06) (119905) = 119879 (08) (119905) =

3

4

if 119905 = 08

1

2

if 119905 = 07

0 if 119905 = 06

119879 (07) (119905) =

0 if 119905 = 08

1

3

if 119905 = 07

3

4

if 119905 = 06

(40)

The 120572-level sets are

[11987906]12

= [11987908]12

= 07 08

[11987906]34

= [11987908]34

= 08

[11987907]34

= 06

[11987907]13

= 07

(41)

Journal of Function Spaces 7

Note that 119865119909[119879119909]

120572(119909)

(119905) = sup(119865119909119910(119905) 119910 isin [119879119909]

120572(119909)) for any

119909 isin 119883 and 119905 isin R and some 120572(119909) isin (0 1] The probabilitydensity functions are as follows with 119865

119909119910(119905) = 119905(119905 + |119909 minus 119910|)

forall119909 119910 isin 119883 forall119905 isin R+

11986506[11987906]

34

(119905) = 1198650608

(119905) =

119905

119905 + 02

forall119905 isin R+

11986506[11987906]

12

(119905) = 1198650607

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

12

(119905) = 1198650807

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

34

(119905) = 1198650808

(119905) = 1 forall119905 isin R+

11986507[11987907]

34

(119905) = 1198650706

(119905) =

119905

119905 + 01

forall119905 isin R+

11986507[11987907]

13

(119905) = 1198650707

(119905) = 1 forall119905 isin R+

119867[119879119909]12[119879119909]12

(119905) =

119905

119905 + 01

for 119909 = 06 08 forall119905 isin R+

119867[119879119909]34[119879119909]34

(119905) = 1 for 119909 = 06 08 forall119905 isin R+

119867[119879119909]12[119879119910]34

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

13

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

34

(119905) =

119905

119905 + 02

for 119909 119910 = 06 08 forall119905 isin R+

119867[11987907]

13[11987907]

13

(119905) = 1 forall119905 isin R+

119867[11987907]

34[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

119867[11987907]

13[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

(42)

Assume that the contractive condition of Theorem 2 holdsunder the form

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905) ge 120572 (119909119899 119909

119899+1 119905 119899)

sdot [119865119909119899[119879119909119899]120572(119909119899)

(119896minus1

119905) + 119865119909119899+1[119879119909119899+1]120572(119909119899+1)

(119896minus1

119905)

+ 119865119909119899119909119899+1

(119896minus1

119905)] 119899 isin Z0+

(43)

for sequences 119909119899 sub 119883 with initial points 119909

0= 119909 isin 119883 119909

1isin

[1198791199090]120572 where 120572(119909 119910 119905 119899) = 120572

119894(119909 119910 119905 119899) 119894 = 1 2 3 forall119905 isin R

+

satisfies

120572 (119909 119910 119905 119899) = 120572119886(119905) le

119896119905

3 (119896119905 + 02)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

120572 (119909 119910 119905 119899) = 120572119887(119905) le

1

3

(1 minus 119890minus120582119905

)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

(44)

for some ℓ(ge 2) isin Z0+ some 119896 isin (0 1) and some 120582 isin R

+

Note that [11987907]13

= 07 with 11986507[11987907]

13

(119905) = 1198650707

(119905) =

1 forall119905 isin R+is a probabilistic 13-fuzzy fixed point of 119879

119883 rarr F(119883) to which the sequences 06 08 07 07 08 06 07 07 converge

3 Further Results

Note that the uniqueness of probabilistic fuzzy fixed points isnot an interesting property to study in the context of prob-abilistic fuzzy fixed point since distinct level sets associatedwith mappings of the form 119879 119883 rarr F(119883) can easily haveintersections of cardinal greater than one in many problemsin the fuzzy context The subsequent result gives conditionsfor the case when several distinct probabilistic fuzzy fixedpoints if they exist are in the intersections of their respectivelevel sets under slightly extended contractive conditions ofthose given in Section 2The extended conditions are of directapplicability

Theorem 7 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 with an

extended contractive condition (12) such that it also holds forany 119909 119910 isin 119883 being probabilistic 120572(119909lowast)-fuzzy fixed points119909 = 119909

lowast

isin [119879119909lowast

]120572(119909lowast)and 119910 = 119910

lowast

isin [119879119910lowast

]120572(119910lowast)of 119879 119883 rarr

F(119883) Then the set [119879119910lowast]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)is nonempty and

119909lowast

119910lowast

isin ([119879119910lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast))

Proof If119909lowast = 119910lowast the result is obvious Assume that there exista probabilistic 120572(119909lowast)-fuzzy fixed point 119909lowast isin [119879119909lowast]

120572(119909lowast)and a

probabilistic 120572(119910lowast)-fuzzy fixed point 119910lowast( = 119909lowast

) isin [119879119910lowast

]120572(119910lowast)

Consider two convergent sequences 119909119899 rarr 119909

lowast and 119910119899 rarr

119910lowast in119883 Then

119865119909119899119910119899

(119905) ge Δ119872(119865

119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

119865119910119899119910lowast (

119905

4

))) forall119905 isin R+

lim inf119899rarrinfin

119865119909119899119910119899

(119905) ge lim inf119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909lowast119910lowast (

119905

4

) 119865119910119899119910lowast (

119905

4

)))

ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

lim119899rarrinfin

119865119910119899119910lowast (

119905

4

))) = Δ119872(1 Δ

119872(119865

119909lowast119910lowast (

119905

4

) 1))

ge 119865119909lowast119910lowast (

119905

4

) forall119905 isin R+

(45)

Assume that 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+

Then from the extended contractive condition (12) and since1 = 119865

119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905) ge 119865119909lowast119910lowast(119905) forall119905 isin R

+

one gets for some 119911119909isin [119879119909

lowast

]120572(119909lowast)and 119911

119910isin [119879119910

lowast

]120572(119910lowast)

8 Journal of Function Spaces

since [119879119909lowast]120572(119909lowast)and [119879119910lowast]

120572(119910lowast)are members of 119862119861(119883) that

is nonempty closed and bounded sets

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905)

= min (119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905))

= min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge 1198861(119909

lowast

119910lowast

119896119905)

sdot 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) + 1198862(119909

lowast

119910lowast

119896119905) 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905)

+ 1198863(119909

lowast

119910lowast

119896119905) 119865119909lowast119910lowast (119905) ge (119886

1(119909

lowast

119910lowast

119896119905)

+ 1198862(119909

lowast

119910lowast

119896119905) + 1198863(119909

lowast

119910lowast

119896119905)) 119865119909lowast119910lowast (119905)

(46)

so that

min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) 119865119909lowast119910lowast (119896

minus1

119905)

ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) Δ119872(119865

119909lowast119911119910

(

119896minus1

119905

2

) 119865119911119910119910lowast (

119896minus1

119905

2

))

ge

119898

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)] Δ119872(119865

119909lowast119911119910

((2119896)minus119898

119905) 119865119911119910119910lowast ((2119896)

119898

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot lim inf119899rarrinfin

Δ119872(119865

119909lowast119911119910

((2119896)minus119899

119905) 119865119911119910119910lowast ((2119896)

minus119899

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot Δ119872(119865

119909lowast119911119910

( lim119899rarrinfin

(2119896)minus119899

119905) 119865119911119910119910lowast ( lim

119899rarrinfin

(2119896)minus119899

119905))

= lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])119867 (+infinminus

) = 1 forall119905 isin R+ forall119898 isin Z

+

(47)

and then 119910lowast = 119911119909 119909lowast = 119911

119910([119879119910

lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)) so that

[119879119910lowast

]120572(119910lowast)cap[119879119909

lowast

]120572(119909lowast)is nonemptyWe can now assume that

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) le 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+ It is direct to

prove in a close way to the above proof that 119910lowast isin [119879119910lowast]120572(119910lowast)cap

[119879119909lowast

]120572(119909lowast)

Remark 8 Note that a direct consequence of Theorem 7from the definition of the level sets is that

119909lowast

119910lowast

isin (( ⋂

120573isin(0120572(119909lowast)]

[119879119909lowast

]120573) cap ( ⋂

120573isin(0120572(119910lowast)]

[119879119910lowast

]120573))

(48)

if 119909lowast isin [119879119909lowast

]120572(119909lowast)and 119910lowast isin [119879119910

lowast

]120572(119910lowast)are any probabilistic

120572(119909lowast

) and 120572(119910lowast)-fuzzy fixed points

Remark 9 Note that corollaries to Theorem 7 can also bestated ldquomutatis-mutandisrdquo under extendedcontractiveconditions

for probabilistic fuzzy fixed points to those given in Corollar-ies 3ndash5

Theorem 10 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 and condi-

tions (1)ndash(3) where the contractive condition

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(49)

is extended to hold for any 119909 119910 isin 119883 with 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Proof Note from Theorem 2 a sequence 119909119899 sub 119883 can be

built being a (convergent) Cauchy sequence such that 119909119899 rarr

Journal of Function Spaces 9

119909lowast 119909

119899+1= 119879119909

119899isin [119879119909

119899]120572(119909119899) forall119899 isin Z

0+for any given 119909

0isin 119883

Then

lim119899rarrinfin

119865119909119899119879119909119899

(119905) = lim119899rarrinfin

119865119909119899119909lowast (119905) = 1 forall119905 isin R

+

lim119899rarrinfin

119865119879119909119899119909lowast (119905) = 1 forall119905 isin R

+

since 119865119879119909119899119909lowast (119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

)) 119865119879119909119899119879119909119899

(

119905

4

))

gt min (1 minus 120582 1 minus 120582 1) = 1 minus 120582

(50)

for any given 119905 isin R+ 120582 isin (0 1) 119899(ge 119873

0) isin Z

0+ and 119873

0=

1198730(120576 120582) ge max(119873

1 119873

2) such that

119865119909lowast119909119899

(

119905

2

) gt 1 minus 120582

for 119899 (ge 1198731) isin Z

0+and some 119873

1= 119873

1(120576 120582) isin Z

0+

119865119909119899+1119909119899

(

119905

4

) gt 1 minus 120582

for 119899 (ge 1198732) isin Z

0+and some 119873

2= 119873

2(120576 120582) isin Z

0+

(51)

since 119865119879119909119899119879119909119899

(1199054) = 1 forall119905 isin R+ forall119899 isin Z

0+from property (1)

of (3) for PM-spaces

Theorem 11 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

(52)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894isin [0 1] for 119894 =

1 3 4 5 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 =

5

sum

119894=1

119886119894le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(53)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(54)

Then a sequence 119909119899 may be constructed for any given

arbitrary 1199091= 119909 isin 119883 such that 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883)

with 120572(119909119899) sube (0 1] satisfying lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isinR+If in addition (119883 F) is endowed with the minimum

triangular norm Δ119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Since [119879119909]120572(119909)

[119879119910]120572(119910)

sub 119883

max (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min119911isin[119879119909]

120572(119909)120596isin[119879119910]

120572(119910)

(119865119911[119879119910]

120572(119910)

(119896119905) 119865120596[119879119909]

120572(119909)

(119896119905))

= 119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905) + 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

forall119905 isin R+ forall119909 119910 isin 119883

(55)

Now for any given 119909 119910 isin 119883 Then the following cases canoccur

(a) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905) forall119905 isin R+

(1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4) 119865

119910[119879119910]120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4+ 119886

5) 119865

119909119910(119905) forall119905 isin R

+

(56)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(57)

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 7

Note that 119865119909[119879119909]

120572(119909)

(119905) = sup(119865119909119910(119905) 119910 isin [119879119909]

120572(119909)) for any

119909 isin 119883 and 119905 isin R and some 120572(119909) isin (0 1] The probabilitydensity functions are as follows with 119865

119909119910(119905) = 119905(119905 + |119909 minus 119910|)

forall119909 119910 isin 119883 forall119905 isin R+

11986506[11987906]

34

(119905) = 1198650608

(119905) =

119905

119905 + 02

forall119905 isin R+

11986506[11987906]

12

(119905) = 1198650607

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

12

(119905) = 1198650807

(119905) =

119905

119905 + 01

forall119905 isin R+

11986508[11987908]

34

(119905) = 1198650808

(119905) = 1 forall119905 isin R+

11986507[11987907]

34

(119905) = 1198650706

(119905) =

119905

119905 + 01

forall119905 isin R+

11986507[11987907]

13

(119905) = 1198650707

(119905) = 1 forall119905 isin R+

119867[119879119909]12[119879119909]12

(119905) =

119905

119905 + 01

for 119909 = 06 08 forall119905 isin R+

119867[119879119909]34[119879119909]34

(119905) = 1 for 119909 = 06 08 forall119905 isin R+

119867[119879119909]12[119879119910]34

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

13

(119905) =

119905

119905 + 01

for 119909 119910 = 06 08 forall119905 isin R+

119867[119879119909]12[11987907]

34

(119905) =

119905

119905 + 02

for 119909 119910 = 06 08 forall119905 isin R+

119867[11987907]

13[11987907]

13

(119905) = 1 forall119905 isin R+

119867[11987907]

34[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

119867[11987907]

13[11987907]

34

(119905) =

119905

119905 + 01

forall119905 isin R+

(42)

Assume that the contractive condition of Theorem 2 holdsunder the form

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119905) ge 120572 (119909119899 119909

119899+1 119905 119899)

sdot [119865119909119899[119879119909119899]120572(119909119899)

(119896minus1

119905) + 119865119909119899+1[119879119909119899+1]120572(119909119899+1)

(119896minus1

119905)

+ 119865119909119899119909119899+1

(119896minus1

119905)] 119899 isin Z0+

(43)

for sequences 119909119899 sub 119883 with initial points 119909

0= 119909 isin 119883 119909

1isin

[1198791199090]120572 where 120572(119909 119910 119905 119899) = 120572

119894(119909 119910 119905 119899) 119894 = 1 2 3 forall119905 isin R

+

satisfies

120572 (119909 119910 119905 119899) = 120572119886(119905) le

119896119905

3 (119896119905 + 02)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

120572 (119909 119910 119905 119899) = 120572119887(119905) le

1

3

(1 minus 119890minus120582119905

)

119894 = 1 2 3 forall119905 isin R+ forall119899 (le ℓ) isin Z

0+

(44)

for some ℓ(ge 2) isin Z0+ some 119896 isin (0 1) and some 120582 isin R

+

Note that [11987907]13

= 07 with 11986507[11987907]

13

(119905) = 1198650707

(119905) =

1 forall119905 isin R+is a probabilistic 13-fuzzy fixed point of 119879

119883 rarr F(119883) to which the sequences 06 08 07 07 08 06 07 07 converge

3 Further Results

Note that the uniqueness of probabilistic fuzzy fixed points isnot an interesting property to study in the context of prob-abilistic fuzzy fixed point since distinct level sets associatedwith mappings of the form 119879 119883 rarr F(119883) can easily haveintersections of cardinal greater than one in many problemsin the fuzzy context The subsequent result gives conditionsfor the case when several distinct probabilistic fuzzy fixedpoints if they exist are in the intersections of their respectivelevel sets under slightly extended contractive conditions ofthose given in Section 2The extended conditions are of directapplicability

Theorem 7 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 with an

extended contractive condition (12) such that it also holds forany 119909 119910 isin 119883 being probabilistic 120572(119909lowast)-fuzzy fixed points119909 = 119909

lowast

isin [119879119909lowast

]120572(119909lowast)and 119910 = 119910

lowast

isin [119879119910lowast

]120572(119910lowast)of 119879 119883 rarr

F(119883) Then the set [119879119910lowast]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)is nonempty and

119909lowast

119910lowast

isin ([119879119910lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast))

Proof If119909lowast = 119910lowast the result is obvious Assume that there exista probabilistic 120572(119909lowast)-fuzzy fixed point 119909lowast isin [119879119909lowast]

120572(119909lowast)and a

probabilistic 120572(119910lowast)-fuzzy fixed point 119910lowast( = 119909lowast

) isin [119879119910lowast

]120572(119910lowast)

Consider two convergent sequences 119909119899 rarr 119909

lowast and 119910119899 rarr

119910lowast in119883 Then

119865119909119899119910119899

(119905) ge Δ119872(119865

119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

119865119910119899119910lowast (

119905

4

))) forall119905 isin R+

lim inf119899rarrinfin

119865119909119899119910119899

(119905) ge lim inf119899rarrinfin

Δ119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909lowast119910lowast (

119905

4

) 119865119910119899119910lowast (

119905

4

)))

ge Δ119872( lim119899rarrinfin

119865119909lowast119909119899

(

119905

2

) Δ119872(119865

119909lowast119910lowast (

119905

4

)

lim119899rarrinfin

119865119910119899119910lowast (

119905

4

))) = Δ119872(1 Δ

119872(119865

119909lowast119910lowast (

119905

4

) 1))

ge 119865119909lowast119910lowast (

119905

4

) forall119905 isin R+

(45)

Assume that 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+

Then from the extended contractive condition (12) and since1 = 119865

119909lowast[119879119909lowast]120572(119909lowast)

(119905) = 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905) ge 119865119909lowast119910lowast(119905) forall119905 isin R

+

one gets for some 119911119909isin [119879119909

lowast

]120572(119909lowast)and 119911

119910isin [119879119910

lowast

]120572(119910lowast)

8 Journal of Function Spaces

since [119879119909lowast]120572(119909lowast)and [119879119910lowast]

120572(119910lowast)are members of 119862119861(119883) that

is nonempty closed and bounded sets

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905)

= min (119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905))

= min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge 1198861(119909

lowast

119910lowast

119896119905)

sdot 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) + 1198862(119909

lowast

119910lowast

119896119905) 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905)

+ 1198863(119909

lowast

119910lowast

119896119905) 119865119909lowast119910lowast (119905) ge (119886

1(119909

lowast

119910lowast

119896119905)

+ 1198862(119909

lowast

119910lowast

119896119905) + 1198863(119909

lowast

119910lowast

119896119905)) 119865119909lowast119910lowast (119905)

(46)

so that

min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) 119865119909lowast119910lowast (119896

minus1

119905)

ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) Δ119872(119865

119909lowast119911119910

(

119896minus1

119905

2

) 119865119911119910119910lowast (

119896minus1

119905

2

))

ge

119898

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)] Δ119872(119865

119909lowast119911119910

((2119896)minus119898

119905) 119865119911119910119910lowast ((2119896)

119898

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot lim inf119899rarrinfin

Δ119872(119865

119909lowast119911119910

((2119896)minus119899

119905) 119865119911119910119910lowast ((2119896)

minus119899

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot Δ119872(119865

119909lowast119911119910

( lim119899rarrinfin

(2119896)minus119899

119905) 119865119911119910119910lowast ( lim

119899rarrinfin

(2119896)minus119899

119905))

= lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])119867 (+infinminus

) = 1 forall119905 isin R+ forall119898 isin Z

+

(47)

and then 119910lowast = 119911119909 119909lowast = 119911

119910([119879119910

lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)) so that

[119879119910lowast

]120572(119910lowast)cap[119879119909

lowast

]120572(119909lowast)is nonemptyWe can now assume that

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) le 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+ It is direct to

prove in a close way to the above proof that 119910lowast isin [119879119910lowast]120572(119910lowast)cap

[119879119909lowast

]120572(119909lowast)

Remark 8 Note that a direct consequence of Theorem 7from the definition of the level sets is that

119909lowast

119910lowast

isin (( ⋂

120573isin(0120572(119909lowast)]

[119879119909lowast

]120573) cap ( ⋂

120573isin(0120572(119910lowast)]

[119879119910lowast

]120573))

(48)

if 119909lowast isin [119879119909lowast

]120572(119909lowast)and 119910lowast isin [119879119910

lowast

]120572(119910lowast)are any probabilistic

120572(119909lowast

) and 120572(119910lowast)-fuzzy fixed points

Remark 9 Note that corollaries to Theorem 7 can also bestated ldquomutatis-mutandisrdquo under extendedcontractiveconditions

for probabilistic fuzzy fixed points to those given in Corollar-ies 3ndash5

Theorem 10 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 and condi-

tions (1)ndash(3) where the contractive condition

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(49)

is extended to hold for any 119909 119910 isin 119883 with 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Proof Note from Theorem 2 a sequence 119909119899 sub 119883 can be

built being a (convergent) Cauchy sequence such that 119909119899 rarr

Journal of Function Spaces 9

119909lowast 119909

119899+1= 119879119909

119899isin [119879119909

119899]120572(119909119899) forall119899 isin Z

0+for any given 119909

0isin 119883

Then

lim119899rarrinfin

119865119909119899119879119909119899

(119905) = lim119899rarrinfin

119865119909119899119909lowast (119905) = 1 forall119905 isin R

+

lim119899rarrinfin

119865119879119909119899119909lowast (119905) = 1 forall119905 isin R

+

since 119865119879119909119899119909lowast (119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

)) 119865119879119909119899119879119909119899

(

119905

4

))

gt min (1 minus 120582 1 minus 120582 1) = 1 minus 120582

(50)

for any given 119905 isin R+ 120582 isin (0 1) 119899(ge 119873

0) isin Z

0+ and 119873

0=

1198730(120576 120582) ge max(119873

1 119873

2) such that

119865119909lowast119909119899

(

119905

2

) gt 1 minus 120582

for 119899 (ge 1198731) isin Z

0+and some 119873

1= 119873

1(120576 120582) isin Z

0+

119865119909119899+1119909119899

(

119905

4

) gt 1 minus 120582

for 119899 (ge 1198732) isin Z

0+and some 119873

2= 119873

2(120576 120582) isin Z

0+

(51)

since 119865119879119909119899119879119909119899

(1199054) = 1 forall119905 isin R+ forall119899 isin Z

0+from property (1)

of (3) for PM-spaces

Theorem 11 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

(52)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894isin [0 1] for 119894 =

1 3 4 5 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 =

5

sum

119894=1

119886119894le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(53)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(54)

Then a sequence 119909119899 may be constructed for any given

arbitrary 1199091= 119909 isin 119883 such that 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883)

with 120572(119909119899) sube (0 1] satisfying lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isinR+If in addition (119883 F) is endowed with the minimum

triangular norm Δ119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Since [119879119909]120572(119909)

[119879119910]120572(119910)

sub 119883

max (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min119911isin[119879119909]

120572(119909)120596isin[119879119910]

120572(119910)

(119865119911[119879119910]

120572(119910)

(119896119905) 119865120596[119879119909]

120572(119909)

(119896119905))

= 119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905) + 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

forall119905 isin R+ forall119909 119910 isin 119883

(55)

Now for any given 119909 119910 isin 119883 Then the following cases canoccur

(a) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905) forall119905 isin R+

(1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4) 119865

119910[119879119910]120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4+ 119886

5) 119865

119909119910(119905) forall119905 isin R

+

(56)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(57)

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Function Spaces

since [119879119909lowast]120572(119909lowast)and [119879119910lowast]

120572(119910lowast)are members of 119862119861(119883) that

is nonempty closed and bounded sets

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) ge 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905)

= min (119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) 119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905))

= min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge 1198861(119909

lowast

119910lowast

119896119905)

sdot 119865119909lowast[119879119909lowast]120572(119909lowast)

(119905) + 1198862(119909

lowast

119910lowast

119896119905) 119865119910lowast[119879119910lowast]120572(119910lowast)

(119905)

+ 1198863(119909

lowast

119910lowast

119896119905) 119865119909lowast119910lowast (119905) ge (119886

1(119909

lowast

119910lowast

119896119905)

+ 1198862(119909

lowast

119910lowast

119896119905) + 1198863(119909

lowast

119910lowast

119896119905)) 119865119909lowast119910lowast (119905)

(46)

so that

min (119865119909lowast119911119910

(119896119905) 119865119910lowast119911119909

(119896119905)) ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) 119865119909lowast119910lowast (119896

minus1

119905)

ge (1198861(119909

lowast

119910lowast

119905) + 1198862(119909

lowast

119910lowast

119905) + 1198863(119909

lowast

119910lowast

119905)) Δ119872(119865

119909lowast119911119910

(

119896minus1

119905

2

) 119865119911119910119910lowast (

119896minus1

119905

2

))

ge

119898

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)] Δ119872(119865

119909lowast119911119910

((2119896)minus119898

119905) 119865119911119910119910lowast ((2119896)

119898

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot lim inf119899rarrinfin

Δ119872(119865

119909lowast119911119910

((2119896)minus119899

119905) 119865119911119910119910lowast ((2119896)

minus119899

119905))

ge lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])

sdot Δ119872(119865

119909lowast119911119910

( lim119899rarrinfin

(2119896)minus119899

119905) 119865119911119910119910lowast ( lim

119899rarrinfin

(2119896)minus119899

119905))

= lim119899rarrinfin

(

119899

prod

119894=1

[1198861(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198862(119909

lowast

119910lowast

119896minus119894+1

119905) + 1198863(119909

lowast

119910lowast

119896minus119894+1

119905)])119867 (+infinminus

) = 1 forall119905 isin R+ forall119898 isin Z

+

(47)

and then 119910lowast = 119911119909 119909lowast = 119911

119910([119879119910

lowast

]120572(119910lowast)cap [119879119909

lowast

]120572(119909lowast)) so that

[119879119910lowast

]120572(119910lowast)cap[119879119909

lowast

]120572(119909lowast)is nonemptyWe can now assume that

119865119910lowast[119879119909lowast]120572(119909lowast)

(119896119905) le 119865119909lowast[119879119910lowast]120572(119910lowast)

(119896119905) forall119905 isin R+ It is direct to

prove in a close way to the above proof that 119910lowast isin [119879119910lowast]120572(119910lowast)cap

[119879119909lowast

]120572(119909lowast)

Remark 8 Note that a direct consequence of Theorem 7from the definition of the level sets is that

119909lowast

119910lowast

isin (( ⋂

120573isin(0120572(119909lowast)]

[119879119909lowast

]120573) cap ( ⋂

120573isin(0120572(119910lowast)]

[119879119910lowast

]120573))

(48)

if 119909lowast isin [119879119909lowast

]120572(119909lowast)and 119910lowast isin [119879119910

lowast

]120572(119910lowast)are any probabilistic

120572(119909lowast

) and 120572(119910lowast)-fuzzy fixed points

Remark 9 Note that corollaries to Theorem 7 can also bestated ldquomutatis-mutandisrdquo under extendedcontractiveconditions

for probabilistic fuzzy fixed points to those given in Corollar-ies 3ndash5

Theorem 10 Consider a complete probabilistic metric space(119883 F Δ

119872) under all the assumptions of Theorem 2 and condi-

tions (1)ndash(3) where the contractive condition

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861(119909 119910 119896119905) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198862(119909 119910 119896119905) 119865

119910[119879119910]120572(119910)

(119905)

+ 1198863(119909 119910 119896119905) 119865

119909119910(119905)

(49)

is extended to hold for any 119909 119910 isin 119883 with 119886119894 119883 times 119883 times R

0+rarr

[0 1] for 119894 = 1 3 1198862 119883 times 119883 times R

0+rarr [0 1)

Proof Note from Theorem 2 a sequence 119909119899 sub 119883 can be

built being a (convergent) Cauchy sequence such that 119909119899 rarr

Journal of Function Spaces 9

119909lowast 119909

119899+1= 119879119909

119899isin [119879119909

119899]120572(119909119899) forall119899 isin Z

0+for any given 119909

0isin 119883

Then

lim119899rarrinfin

119865119909119899119879119909119899

(119905) = lim119899rarrinfin

119865119909119899119909lowast (119905) = 1 forall119905 isin R

+

lim119899rarrinfin

119865119879119909119899119909lowast (119905) = 1 forall119905 isin R

+

since 119865119879119909119899119909lowast (119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

)) 119865119879119909119899119879119909119899

(

119905

4

))

gt min (1 minus 120582 1 minus 120582 1) = 1 minus 120582

(50)

for any given 119905 isin R+ 120582 isin (0 1) 119899(ge 119873

0) isin Z

0+ and 119873

0=

1198730(120576 120582) ge max(119873

1 119873

2) such that

119865119909lowast119909119899

(

119905

2

) gt 1 minus 120582

for 119899 (ge 1198731) isin Z

0+and some 119873

1= 119873

1(120576 120582) isin Z

0+

119865119909119899+1119909119899

(

119905

4

) gt 1 minus 120582

for 119899 (ge 1198732) isin Z

0+and some 119873

2= 119873

2(120576 120582) isin Z

0+

(51)

since 119865119879119909119899119879119909119899

(1199054) = 1 forall119905 isin R+ forall119899 isin Z

0+from property (1)

of (3) for PM-spaces

Theorem 11 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

(52)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894isin [0 1] for 119894 =

1 3 4 5 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 =

5

sum

119894=1

119886119894le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(53)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(54)

Then a sequence 119909119899 may be constructed for any given

arbitrary 1199091= 119909 isin 119883 such that 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883)

with 120572(119909119899) sube (0 1] satisfying lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isinR+If in addition (119883 F) is endowed with the minimum

triangular norm Δ119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Since [119879119909]120572(119909)

[119879119910]120572(119910)

sub 119883

max (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min119911isin[119879119909]

120572(119909)120596isin[119879119910]

120572(119910)

(119865119911[119879119910]

120572(119910)

(119896119905) 119865120596[119879119909]

120572(119909)

(119896119905))

= 119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905) + 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

forall119905 isin R+ forall119909 119910 isin 119883

(55)

Now for any given 119909 119910 isin 119883 Then the following cases canoccur

(a) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905) forall119905 isin R+

(1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4) 119865

119910[119879119910]120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4+ 119886

5) 119865

119909119910(119905) forall119905 isin R

+

(56)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(57)

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 9

119909lowast 119909

119899+1= 119879119909

119899isin [119879119909

119899]120572(119909119899) forall119899 isin Z

0+for any given 119909

0isin 119883

Then

lim119899rarrinfin

119865119909119899119879119909119899

(119905) = lim119899rarrinfin

119865119909119899119909lowast (119905) = 1 forall119905 isin R

+

lim119899rarrinfin

119865119879119909119899119909lowast (119905) = 1 forall119905 isin R

+

since 119865119879119909119899119909lowast (119905) ge Δ

119872(119865

119909lowast119909119899

(

119905

2

)

Δ119872(119865

119909119899119909119899+1

(

119905

4

)) 119865119879119909119899119879119909119899

(

119905

4

))

gt min (1 minus 120582 1 minus 120582 1) = 1 minus 120582

(50)

for any given 119905 isin R+ 120582 isin (0 1) 119899(ge 119873

0) isin Z

0+ and 119873

0=

1198730(120576 120582) ge max(119873

1 119873

2) such that

119865119909lowast119909119899

(

119905

2

) gt 1 minus 120582

for 119899 (ge 1198731) isin Z

0+and some 119873

1= 119873

1(120576 120582) isin Z

0+

119865119909119899+1119909119899

(

119905

4

) gt 1 minus 120582

for 119899 (ge 1198732) isin Z

0+and some 119873

2= 119873

2(120576 120582) isin Z

0+

(51)

since 119865119879119909119899119879119909119899

(1199054) = 1 forall119905 isin R+ forall119899 isin Z

0+from property (1)

of (3) for PM-spaces

Theorem 11 Let F(119883) be the collection of all fuzzy sets in aPM-space (119883 F) where 119883 is a nonempty abstract set and let119879 119883 rarr F(119883) be a fuzzy mapping Assume that the followingconditions are fulfilled

(1) for each 119909 isin 119883 there exists 120572(119909) isin (0 1] such that[119879119909]

120572(119909)is a nonempty closed bounded subset of 119883

and each sequence 119909119899 sub 119883 of the form 119909

1isin 119883

119909119899+1

isin [119879119909119899]120572(119909119899)with [119879119909

119899]120572(119909119899)isin 119862119861(119883) forall119899 isin Z

0+

which satisfies the following contractive constraint forsome real constant 119896 isin (0 1)

119867[119879119909]120572(119909)

[119879119910]120572(119910)

(119896119905) ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

(52)

forall119909 isin 119883 forall119910 isin [119879119909]120572(119909)

where 119886119894isin [0 1] for 119894 =

1 3 4 5 1198862 119883 times 119883 times R

0+rarr [0 1)

(2)

0 lt 119886 =

5

sum

119894=1

119886119894le 1

forall119905 isin R0+ forall119909 isin 119883 forall119910 isin [119879119909]

120572(119909)sub 119862119861 (119883)

(53)

(3)

lim119873rarrinfin

119873

prod

119894=0

[

1198861(119909

119894+1 119909

119894+2 119896

minus119894+1

119905) + 1198863(119909

119894+1 119909

119894+2 119896

minus119894+1

119905)

1 minus 1198862(119909

119894+1 119909

119894+2 119896

minus119894+1119905)

]

= 1 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861 (119883) forall119905 isin R

+ forall119899 isin Z

0+

(54)

Then a sequence 119909119899 may be constructed for any given

arbitrary 1199091= 119909 isin 119883 such that 119909

119899+1isin [119879119909

119899]120572(119909119899)isin 119862119861(119883)

with 120572(119909119899) sube (0 1] satisfying lim

119899rarrinfin119865119909119899119909119899+1

(119905) = 1 forall119905 isinR+If in addition (119883 F) is endowed with the minimum

triangular norm Δ119872 [0 1] times [0 1] rarr [0 1] and (119883 F Δ

119872)

is a complete Menger space then each of such sequences 119909119899 is

a Cauchy sequence which is convergent to a probabilistic 120572lowast-fuzzy fixed point 119909lowast of 119879 119883 rarr F(119883)

Proof Since [119879119909]120572(119909)

[119879119910]120572(119910)

sub 119883

max (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min (119865119909[119879119910]

120572(119910)

(119896119905) 119865119910[119879119909]

120572(119909)

(119896119905))

ge min119911isin[119879119909]

120572(119909)120596isin[119879119910]

120572(119910)

(119865119911[119879119910]

120572(119910)

(119896119905) 119865120596[119879119909]

120572(119909)

(119896119905))

= 119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905) + 1198863119865119910[119879119909]

120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

forall119905 isin R+ forall119909 119910 isin 119883

(55)

Now for any given 119909 119910 isin 119883 Then the following cases canoccur

(a) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905) forall119905 isin R+

(1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119896119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4) 119865

119910[119879119910]120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge (1198861+ 119886

4+ 119886

5) 119865

119909119910(119905) forall119905 isin R

+

(56)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(57)

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Function Spaces

(b) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119905) ge 119865119910[119879119909]

120572(119909)

(119905)

ge (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ (1198861+ 119886

4+ 119886

5) 119865

119910[119879119910]120572(119910)

(119905)

forall119905 isin R+

(58)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(59)

(c) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

(1 minus 1198862minus 119886

3) 119865

119909[119879119910]120572(119910)

(119905) ge (1 minus 1198862minus 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

ge 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905)

ge max [(1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) 1198861119865119909[119879119909]

120572(119909)

(119905) + (1198864+ 119886

5) 119865

119909119910(119905)]

ge (1198861+ 119886

4+ 119886

5)min (119865

119909[119879119909]120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(60)

so that

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(61)

(d) If 119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge (1198861+ 119886

4) 119865

119909[119879119909]120572(119909)

(119905) + (1198862+ 119886

3) 119865

119910[119879119909]120572(119909)

(119905)

+ 1198865119865119909119910

(119905) forall119905 isin R+

119865119909[119879119910]

120572(119910)

(119896119905) ge 119865119910[119879119909]

120572(119909)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(62)

(e) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909119910

(119905)

forall119905 isin R+

(63)

(f) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) ge

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) le 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119905) ge 119865119909[119879119910]

120572(119910)

(119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119910[119879119910]

120572(119910)

(119905) forall119905 isin R+

(64)

(g) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119909119910

(119905))

forall119905 isin R+

(65)

(h) If 119865119909[119879119910]

120572(119910)

(119896119905) le 119865119910[119879119909]

120572(119909)

(119896119905) 119865119909[119879119909]

120572(119909)

(119905) le

119865119910[119879119910]

120572(119910)

(119905) and 119865119910[119879119910]

120572(119910)

(119905) ge 119865119909119910(119905) forall119905 isin R

+then

119865119910[119879119909]

120572(119909)

(119896119905) ge 119865119909[119879119910]

120572(119910)

(119896119905)

ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

min (119865119909[119879119909]

120572(119909)

(119905) 119865119910[119879119910]

120572(119910)

(119905))

=

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

119865119909[119879119909]

120572(119909)

(119905) forall119905 isin R+

(66)

Now take 1199091isin 119883 119909

2= 119910 isin [119879119909]

120572(119909) and any 119911 = 119909

3isin

[119879119910]120572(119910)

such that119865119909119910(119905) = 119865

119909[119879119909](119905)Then one gets for Cases

(a)ndash(h)

11986511990921199093

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905) = 11986511990911199092

(119896minus1

119905) (67)

and we can construct a sequence 119909119899 sub 119883 with 119909

1= 119909 isin

119883 arbitrary 119909119899+1

isin [119879119909119899]120572(119909119899)isin 119862119861(119883) for some sequence

120572(119909119899) sub (0 1] such that since (119886

1+119886

4+119886

5)(1minus119886

2minus119886

3) = 1

one gets proceeding recursively

119865119909119899+1119909119899

(119905) ge

1198861+ 119886

4+ 119886

5

1 minus 1198862minus 119886

3

11986511990911199092

(119896minus1

119905)

ge 11986511990911199092

(119896minus119899+1

119905)

forall119899 (ge 2) isin Z+ forall119905 isin R

+

(68)

so that lim119899rarrinfin

119865119909119899+1119909119899

(119905) = 1 forall119905 isin R+

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 11

Remark 12 Extensions of Theorem 11 are direct for the casewhen the subsequent contractive condition holds instead of(48)

119867(119865[119879119909]120572(119909)

(119896119905) [119879119910]120572(119910)

(119896119905))

ge min (1 1198861119865119909[119879119909]

120572(119909)

(119905) + 1198862119865119909[119879119910]

120572(119910)

(119905)

+ 1198863119865119910[119879119909]

120572(119909)

(119905) + 1198864119865119910[119879119910]

120572(119910)

(119905) + 1198865119865119909119910

(119905))

forall119909 119910 isin 119883 forall119905 isin R+

(69)

Extensions of Theorem 11 and its variant of Remark 12 to thelight of Theorems 2 and 10 are direct concerning the casewhen the coefficients of the respective contractive conditionare functions 119886

119894 119883 times 119883 times R

0+rarr [0 1] for 119894 = 1 3 4 5

1198862 119883times119883timesR

0+rarr [0 1) Also close examples to Example 6

can be given for the more general contractive conditions ofthis section

Conflict of Interests

The author declares that he has no conflict of interests

Acknowledgments

The author is very grateful to the Spanish Government forGrant DPI2012-30651 and to the Basque Government andUPVEHU for Grants IT378-10 SAIOTEK S-PE13UN039and UFI 201107

References

[1] S Heilpern ldquoFuzzy mappings and fixed point theoremrdquo Journalof Mathematical Analysis and Applications vol 83 no 2 pp566ndash569 1981

[2] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1 pp59ndash62 2014

[3] A Chitra and P V Subrahmanyam ldquoFuzzy sets and fixedpointsrdquo Journal of Mathematical Analysis and Applications vol124 no 2 pp 584ndash590 1987

[4] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[5] B Singh S Jain and S Jain ldquoGeneralized theorems on fuzzymetric spacesrdquo Southeast Asian Bulletin of Mathematics vol 31no 5 pp 963ndash978 2007

[6] A Azam and I Beg ldquoCommon fuzzy fixed points for fuzzymappingsrdquo Fixed Point Theory and Applications vol 2013article 14 2013

[7] Y J Cho H K Pathak S M Kang and J S Jung ldquoCommonfixed points of compatible maps of type (120573) on fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 93 no 1 pp 99ndash111 1998

[8] D Qiu and L Shu ldquoSupremum metric on the space of fuzzysets and common fixed point theorems for fuzzy mappingsrdquoInformation Sciences vol 178 no 18 pp 3595ndash3604 2008

[9] M Abbas I Altun and D Gopal ldquoCommon fixed pointtheorems for non compatible mappings in fuzzy metric spacesrdquo

Bulletin of Mathematical Analysis and Applications vol 1 no 2pp 47ndash56 2009

[10] S PhiangsungnoenW Sintunavarat and P Kumam ldquoCommon120572-fuzzy fixed point theorems for fuzzy mappings via 120573

119865-

admissible pairrdquo Journal of Intelligent and Fuzzy Systems vol 27no 5 pp 2463ndash2472 2014

[11] A Jain A Sharma V Gupta and A Tiwari ldquoCommon fixedpoint theorem in fuzzy metric space with special reference tooccasionally weakly compatible mappingsrdquo Journal of Mathe-matics and Computer Science vol 4 no 2 pp 374ndash383 2014

[12] D Turkoglu and B E Rhoades ldquoA fixed fuzzy point for fuzzymapping in complete metric spacesrdquo Mathematical Communi-cations vol 10 pp 115ndash121 2005

[13] K Singh Sisodia D Singh and M S Rathore ldquoA commonfixed point theorem for subcompatibility and occasionally weakcompatibility in intuitionistic fuzzy metric spacesrdquo GeneralMathematics Notes vol 21 no 1 pp 73ndash85 2014

[14] S Manro H Bouharjera and S Singh ldquoA common fixedpoint theorem in intuitionistic fuzzy metric space by usingsub-compatible mapsrdquo International Journal of ContemporaryMathematical Sciences vol 5 no 55 pp 2699ndash2707 2010

[15] M De la Sen and A Ibeas ldquoProperties of convergence of a classof iterative processes generated by sequences of self-mappingswith applications to switched dynamic systemsrdquo Journal ofInequalities and Applications vol 2014 article 498 2014

[16] M De la Sen and E Karapinar ldquoOn a cyclic Jungck modifiedTS-iterative procedure with application examplesrdquo AppliedMathematics and Computation vol 233 pp 383ndash397 2014

[17] M De la Sen ldquoAbout robust stability of caputo linear fractionaldynamic systems with time delays through fixed point theoryrdquoFixed Point Theory and Applications vol 2011 Article ID867932 2011

[18] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[19] S Karpagam and S Agrawal ldquoBest proximity point theoremsfor p-cyclic Meir-Keeler contractionsrdquo Fixed Point Theory andApplications vol 2009 article 197308 9 pages 2009

[20] M De la Sen ldquoLinking contractive self-mappings and cyclicmeir-keeler contractions with kannan self-mappingsrdquo FixedPointTheory andApplications vol 2010 Article ID 572057 2010

[21] C Di Bari T Suzuki and C Vetro ldquoBest proximity pointsfor cyclicMeir-Keeler contractionsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 69 no 11 pp 3790ndash3794 2008

[22] S Rezapour M Derafshpour and N Shahzad ldquoOn the exis-tence of best proximity points of cyclic contractionsrdquo Advancesin Dynamical Systems and Applications vol 6 no 1 pp 33ndash402011

[23] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 70 no 10 pp 3665ndash3671 2009

[24] M Derafshpour S Rezapour and N Shahzad ldquoBest proximitypoints of cyclic 120593-contractions on reflexive Banach spacesrdquoFixed Point Theory and Applications vol 2010 Article ID946178 7 pages 2010

[25] B S Choudhury KDas and SK Bhandari ldquoCyclic contractionresult in 2-Menger spacerdquo Bulletin of International Mathemati-cal Virtual Institute vol 2 no 1 pp 223ndash234 2012

[26] I Beg and M Abbas ldquoFixed points and best approximation inmenger convex metric spacesrdquo Archivum Mathematicum vol41 no 4 pp 389ndash397 2005

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Function Spaces

[27] I Beg A Latif and M Abbas ldquoCoupled fixed points of mixedmonotone operators on probabilistic Banach spacesrdquo ArchivumMathematicum vol 37 no 1 pp 1ndash8 2001

[28] M De la Sen and E Karapınar ldquoSome results on best proximitypoints of cyclic contractions in probabilistic metric spacesrdquoJournal of Function Spaces vol 2015 Article ID 470574 11 pages2015

[29] M De la Sen R P Agarwal and A Ibeas ldquoResults on proximaland generalized weak proximal contractions including the caseof iteration-dependent range setsrdquo Fixed Point Theory andApplications vol 2014 article 169 2014

[30] M Gabeleh ldquoBest proximity point theorems via proximal nonself-mappingrdquo Journal of OptimizationTheory and Applicationsvol 164 no 2 pp 565ndash576 2015

[31] L X Wang Adaptive Fuzzy Systems and Control Design andStability Analysis Prentice-Hall Englewood Cliffs NJ USA1994

[32] M De la Sen A F Roldan and R P Agarwal ldquoOn contractivecyclic fuzzy maps in metric spaces and some related results onfuzzy best proximity points and fuzzy fixed pointsrdquo Fixed PointTheory and Applications vol 2015 article 103 2015

[33] E Pap O Hadzic and R Mesiar ldquoA fixed point theoremin probabilistic metric spaces and an applicationrdquo Journal ofMathematical Analysis and Applications vol 202 no 2 pp 433ndash449 1996

[34] V M Sehgal and A T Bharucha-Reid ldquoFixed points ofcontraction mappings on probabilistic metric spacesrdquoTheory ofComputing Systems vol 6 no 1 pp 97ndash102 1972

[35] M De la Sen S Alonso-Quesada and A Ibeas ldquoOn theasymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraintsrdquo IMA Journal ofMathematical Control and Information vol 32 no 2 pp 359ndash386 2015

[36] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Publishing Amsterdam The Netherlands 1983

[37] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 no 1 pp 313ndash334 1960

[38] M De la Sen andA Ibeas ldquoOn the global stability of an iterativescheme in a probabilistic Menger spacerdquo Journal of Inequalitiesand Applications vol 2015 article 243 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of