Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 967132 9 pageshttpdxdoiorg1011552013967132

Research ArticleSome Fixed Point Theorems in 119887-Metric SpaceEndowed with Graph

Maria Samreen1 Tayyab Kamran2 and Naseer Shahzad3

1 Centre for Advanced Mathematics and Physics National University of Sciences and Technology Sector H-12 Islamabad Pakistan2Department of Mathematics Quaid-i-Azam University Islamabad Pakistan3Department of Mathematics King Abdulaziz University Jeddah Saudi Arabia

Correspondence should be addressed to Naseer Shahzad nshahzadkauedusa

Received 6 July 2013 Revised 5 September 2013 Accepted 6 September 2013

Academic Editor Douglas Anderson

Copyright copy 2013 Maria Samreen et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We define some notions of contraction mappings in 119887-metric space endowed with a graph119866 and subsequently establish some fixedpoint results for such classes of contractions According to the applications of our results we obtain fixed point theorems for cyclicoperators and an existence theorem for the solution of an integral equation

1 Introduction and Preliminaries

The study of 119887-metric spaces was initiated in some worksof Bakhtin Heinonen Bourbaki and Czerwik [1ndash4] After-wards several articles which deal with fixed point theoremsfor single-valued andmultivalued functions in 119887-metric spaceappeared [1ndash8]

Definition 1 (see [1 4]) Let119883 be a set and let 119904 ge 1 be a givenreal number A function 119889 119883 times 119883 rarr R+ is said to be a119887-metric on 119883 and the pair (119883 119889) is called a 119887-metric spaceif for all 119909 119910 119911 isin 119883

(d1) 119889(119909 119910) = 0 if and only if 119909 = 119910

(d2) 119889(119909 119910) = 119889(119910 119909)

(d3) 119889(119909 119911) le 119904[119889(119909 119910) + 119889(119910 119911)]

Note that the class of 119887-metric spaces contains the class ofmetric spaces

Example 2 Let 119883 = 119897119901(R) with 0 lt 119901 lt 1 where 119897

119901(R) =

119909 = 119909119899 sub R sum

infin

119899=1|119909119899|119901

lt infinThen 119889(119909 119910) = (suminfin

119899=1|119909119899minus

119910119899|119901)1119901 is a 119887-metric on 119883 with 119904 = 2

1119901

A sequence 119909119899 in a 119887-metric space 119883 is said to be

convergent if and only if there exists 119909 isin 119883 such that119889(119909119899 119909) rarr 0 as 119899 rarr infin In this case we write lim

119899rarrinfin119909119899

=

119909 A sequence 119909119899 in a 119887-metric space119883 is said to be Cauchy

if and only if 119889(119909119899 119909119898) rarr 0 as119898 119899 rarr infin A 119887-metric space

(119883 119889) is complete if every Cauchy sequence in 119883 convergesIn general a 119887-metric is not continuous

The famous Banach contraction principle [9] infers thatevery contraction on a complete metric space has a uniquefixed point Recently Jachymski [10] introduced the notionof Banach 119866-contraction to generalize Banach contractionprinciple as follows Let (119872 120575) be a metric space let Δ bethe diagonal of the Cartesian product 119872 times 119872 and let 119866 be adirected graph such that the set 119881(119866) of its vertices coincideswith119872 and the set119864(119866) of its edges contains all loops that is119864(119866) supe Δ Assume that 119866 has no parallel edges A mapping119891 119872 rarr 119872 is called a Banach 119866-contraction if (i) forall 119909 119910 isin 119883 ((119909 119910) isin 119864(119866) rArr (119891119909 119891119910) isin 119864(119866)) (ii) exist1205720 lt 120572 lt 1 such that for all 119909 119910 isin 119883 (119909 119910) isin 119864(119866) rArr

120575(119891119909 119891119910) le 120572120575(119909 119910) A mapping 119891 119872 rarr 119872 is knownas Picard operator [11] if 119891 has a unique fixed point 119909

lowast andlim119899rarrinfin

119891119899119909 = 119909

lowast for all 119909 isin 119872Various generalizations of Banachrsquos principle have been

obtained by weakening contractive conditions In this con-text Matkowski [12] introduced class of 120593-contractions in

2 Abstract and Applied Analysis

metric fixed point theory and subsequently further studywas developed in this setting by different authors whenunderlying space was taken to be a partially ordered set (seeeg [13 14])

Let 120593 R+ rarr R+ Consider the following properties

(i)120593

1199051le 1199052rArr 120593(119905

1) le 120593(119905

2) for all 119905

1 1199052isin R+

(ii)120593

120593(119905) lt 119905 for 119905 gt 0

(iii)120593

120593(0) = 0

(iv)120593lim119899rarrinfin

120593119899(119905) = 0 for all 119905 ge 0

(v)120593

suminfin

119899=0120593119899(119905) converges for all 119905 gt 0

It is easily seen that (i)120593and (iv)

120593imply (ii)

120593and (i)

120593and (ii)

120593

imply (iii)120593

We recall that a function 120593 satisfying (i)120593and (iv)

120593is said

to be a comparison function A function 120593 satisfying (i)120593and

(v)120593is known as (119888)-comparison functionAny (119888)-comparison function is a comparison function

but converse may not be true For example 120593(119905) = 119905(1 + 119905)119905 isin R+ is a comparison function but not a (119888)-comparisonfunction On the other hand define 120593(119905) = 1199052 0 le 119905 le 1

and 120593(119905) = 119905 minus (12) 119905 gt 1 and then 120593 is a (119888)-comparisonfunction For details on 120593 contractions we refer the readersto [15 16]Berinde [17] took further step to investigate 120593 contractionswhen the framework was taken to be a 119887-metric space andfor some technical reasons he had to introduce the notionof 119887-comparison function in particular he obtained someestimations for rate of convergence [17] For other relatedresults see also [5 7 17ndash21]

Definition 3 Let 119904 ge 1 be a fixed real number A function 120593

R+ rarr R+ is known as 119887-comparison function if it satisfies(i)120593 and the following holds

(vi)120593

suminfin

119899=0119904119899120593119899(119905) converges for all 119905 isin R+

The concept of 119887-comparison function coincides withcomparison function when 119904 = 1 Let (119883 119889) be a 119887-metricspace with coefficient 119904 ge 1 and then 120593(119905) = 119886119905 119905 isin R+ with0 lt 119886 lt (1119904) is a 119887-comparison function

2 Main Results

Throughout this section let (119883 119889) be a 119887-metric space withcoefficient 119904 ge 1 and Δ is the diagonal of the Cartesianproduct 119883 times 119883 119866 is a directed graph such that the set 119881(119866)

of its vertices coincides with 119883 and the set 119864(119866) of its edgescontains all loops that is 119864(119866) supe Δ Assume that 119866 has noparallel edges We assign to each edge having vertices 119909 and119910 a unique element 119889(119909 119910) Now we introduce the followingdefinition

Definition 4 One says that a mapping 119891 119883 rarr 119883 is a 119887-(120593 119866) contraction if for all 119909 119910 isin 119883

(119891119909 119891119910) isin 119864 (119866) whenever (119909 119910) isin 119864 (119866) (1)

119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) whenever (119909 119910) isin 119864 (119866)

(2)

where 120593 R+ rarr R+ is a comparison function

Remark 5 Note that a Banach 119866-contraction is a 119887-(120593 119866)

contraction

Example 6 Any constant mapping 119891 119883 rarr 119883 is a 119887-(120593 119866)

contraction for any graph 119866 with 119881(119866) = 119883

Example 7 Any self-mapping 119891 on 119883 is trivially a 119887-(120593 1198661)

contraction where 1198661= (119881(119866) 119864(119866)) = (119883 Δ)

Example 8 Let 119883 = R and define 119889 119883 times 119883 rarr R by119889(119909 119910) = |119909 minus 119910|

2 Then 119889 is a 119887-metric on 119883 with 119904 = 2Further 119891119909 = 1199092 for all 119909 isin 119883 Then 119891 is a 119887-(120593 119866

0)

contraction with 120593(119905) = 1199054 and 1198660= (119883119883times119883) Note that 119889

is not a metric on 119883

Definition 9 Two sequences 119909119899 and 119910

119899 in119883 are said to be

equivalent if lim119899rarrinfin

119889(119909119899 119910119899) = 0 and if each of them is a

Cauchy sequence then they are called Cauchy equivalent

As a consequence of Definition 9 we get the followinglemma

Remark 10 Let 119909119899 and 119910

119899 be equivalent sequences in 119883

(i) If 119909119899 converges to 119909 then 119910

119899 also converges to 119909 and

vice versa (ii) 119910119899 is a Cauchy sequence whenever 119909

119899 is a

Cauchy sequence and vice versa

Now we recollect some preliminaries from graph theorywhich we need for the sequel Let 119866 = (119881(119866) and let 119864(119866))

be a directed graph By letter 119866 we denote the undirectedgraph obtained from 119866 by ignoring the direction of edgesIf 119909 and 119910 are vertices in a graph 119866 then a path in 119866 from119909 to 119910 of length 119897 is a sequence 119909

119894119897

119894=0of 119897 + 1 vertices such

that 1199090

= 119909 119909119897

= 119910 and (119909119894minus1

119909119894) isin 119864(119866) for 119894 = 1 119897

A graph 119866 is called connected if there is a path between anytwo vertices 119866 is weakly connected if 119866 is connected For agraph 119866 such that 119864(119866) is symmetric and 119909 is a vertex in 119866the subgraph 119866

119909consisting of all edges and vertices which

are contained in some path beginning at 119909 is known as acomponent of 119866 containing 119909 So that 119881(119866

119909) = [119909]

119866 where

[119909]119866is the equivalence class of a relation 119877 defined on 119881(119866)

by the rule 119910119877119911 if there is a path in119866 from 119910 to 119911 Clearly119866119909

is connected A graph119866 is known as (119862)-graph in119883 [22] if forany sequence 119909

119899 in 119883 with 119909

119899rarr 119909 and (119909

119899 119909119899+1

) isin 119864(119866)

for 119899 isin N then there exists a subsequence 119909119899119896

of 119909119899 such

that (119909119899119896

119909) isin 119864(119866) for 119896 isin N

Proposition 11 Let 119891 119883 rarr 119883 be a 119887-(120593 119866) contractionwhere 120593 R+ rarr R+ is a comparison function then

(i) 119891 is a 119887-(120593 119866) contraction and a 119887-(120593 119866minus1

) contractionas well

(ii) [1199090]119866is 119891-invariant and 119891|

[1199090]119866

is a 119887-(120593 119866119909) contrac-

tion provided that 1199090isin 119883 is such that 119891119909

0isin [1199090]119866

Abstract and Applied Analysis 3

Proof (i) By using (d2) (Definition 1) it can be easily proved(ii) Let 119909 isin [119909

0]119866Then there is a path 119909 = 119911

0 1199111 119911

119897=

1199090between 119909 and 119909

0 Since 119891 is a 119887-(120593 119866) contraction then

(119891119911119894minus1

119891119911119894) isin 119864(119866) for all 119894 = 1 2 119897 Thus 119891119909 isin [119891119909

0]119866

=

[1199090]119866

Suppose (119909 119910) isin 119864(1198661199090

) Then (119891119909 119891119910) isin 119864(119866) since119891 is a 119887-(120593 119866) contraction But [119909

0]119866is 119891 invariant so we

conclude that (119891119909 119891119910) isin 119864(1198661199090

) Condition (2) is satisfiedautomatically as 119866

1199090is a subgraph of 119866

Fromnowon we assume that coefficient of 119887-comparisonfunction is at least as large as the coefficient of 119887-metric 119904

Lemma 12 Let 119891 119883 rarr 119883 be a 119887-(120593 119866) contraction where120593 R+ rarr R+ is a 119887-comparison function Then given any119909 isin 119883 and 119910 isin [119909]

119866 two sequences 119891

119899119909 and 119891

119899119910 are

equivalent

Proof Let 119909 isin 119883 and let 119910 isin [119909]119866 then there exists a path

119909119894119897

119894=0in 119866 from 119909 to 119910 with 119909

0= 119909 119909

119897= 119910 and (119909

119894minus1 119909119894) isin

119864(119866) From Proposition 11 119891 is a 119887-(120593 119866) contraction So

(119891119899

119909119894minus1

119891119899

119909119894) isin 119864 (119866) implies 119889 (119891

119899

119909119894minus1

119891119899

119909119894)

le 120593 (119889 (119891119899minus1

119909119894minus1

119891119899minus1

119909119894))

(3)

for all 119899 isin N and 119894 = 0 1 2 119897 Hence

119889 (119891119899

119909119894minus1

119891119899

119909119894) le 120593119899

(119889 (119909119894minus1

119909119894))

forall119899 isin N 119894 = 0 1 2 119897

(4)

We observe that 119891119899119909119894119897

119894=0is a path in 119866 from 119891

119899119909 to 119891

119899119910

Using (d3) Definition 1 and (4) we have

119889 (119891119899

119909 119891119899

119910) le

119897

sum

119894=1

119904119894

119889 (119891119899

119909119894minus1

119891119899

119909119894)

le

119897

sum

119894=1

119904119894

120593119899

(119889 (119909119894minus1

119909119894))

(5)

Letting 119899 rarr infin we obtain 119889(119891119899119909 119891119899119910) rarr 0

Proposition 13 Let 119891 be a 119887-(120593 119866) contraction where 120593

R+ rarr R+ is a 119887-comparison function Suppose that there is 1199110

in 119883 such that 1198911199110isin [1199110]119866 Then 119891119899119911

0 is a Cauchy sequence

in 119883

Proof Since 1198911199110

isin [1199110]119866 let 119910

119894119903

119894=0be a path from 119911

0to 1198911199110

Then using the same arguments as in Lemma 12 we arrive at

119889 (119891119899

1199110 119891119899+1

1199110) le

119903

sum

119894=1

119904119894

120593119899

(119889 (119910119894minus1

119910119894)) forall119899 isin N

(6)

Let 119898 gt 119899 ge 1 and then from above inequality it follows for119901 ge 1

119889 (119891119899

1199110 119891119899+119901

1199110)

le 119904119889 (119891119899

1199110 119891119899+1

1199110) + 1199042

119889 (119891119899+1

1199110 119891119899+2

1199110)

+ sdot sdot sdot + 119904119901

119889 (119891119899+119901minus1

1199110 119891119899+119901

1199110)

le1

119904119899minus1[

[

119899+119901minus1

sum

119895=119899

119904119895

119889 (119891119895

1199110 119891119895+1

1199110)]

]

le1

119904119899minus1[

[

119903

sum

119894=1

119904119894

119899+119901minus1

sum

119895=119899

119904119895

120593119895

(119889 (119910119894minus1

119910119894))]

]

(7)

Denoting for each 119894 = 1 2 119903

119878119894

119899=

119899

sum

119896=0

119904119896

120593119896

(119889 (119910119894minus1

119910119894)) 119899 ge 1 (8)

relation (7) becomes

119889 (119891119899

1199110 119891119899+119901

1199110) le

1

119904119899minus1[

119903

sum

119894=1

119904119894

[119878119894

119899+119901minus1minus 119878119894

119899minus1]] (9)

since 120593 is a 119887-comparison function so that for each 119894 =

1 2 119903infin

sum

119896=0

119904119896

120593119896

(119889 (119910119894minus1

119910119894)) lt infin (10)

Then corresponding to each 119894 there is a real number 119878119894 such

that

lim119899rarrinfin

119878119894

119899= 119878119894

(11)

In view of (11) relation (9) gives 119889(1198911198991199110 119891119899+119901

1199110) rarr 0 as

119899 rarr infin Which shows that 1198911198991199110 is a Cauchy sequence in

119883

Definition 14 Let 119891 119883 rarr 119883 and let 119910 isin 119883 andthe sequence 119891

119899119910 in 119883 is such that 119891

119899119910 rarr 119909

lowast with(119891119899119910 119891119899+1

119910) isin 119864(119866) for 119899 isin N One says that a graph 119866 is(119862119891)-graph if there exists a subsequence 119891

119899119896119910 and a naturalnumber 119901 such that (119891119899119896119910 119909

lowast) isin 119864(119866) for all 119896 ge 119901 [23] One

says that a graph 119866 is (119867119891)-graph if 119891

119899119910 isin [119909

lowast]119866for 119899 ge 1

then 119903(119891119899119910 119909lowast) rarr 0 (as 119899 rarr infin) where 119903(119891

119899119910 119909lowast) =

sum119872

119894=1119904119894119889(119911119894minus1

119911119894) 119911119894119872

119894=0is a path from 119891

119899119910 to 119909

lowast in 119866

Obviously every (119862)-graph is a (119862119891)-graph for any self-

mapping 119891 on 119883 but converse may not hold as shown in thefollowing

Example 15 Let119883 = [0 1] with respect to 119887-metric 119889(119909 119910) =

|119909 minus 119910|2 Consider a graph 119866 consisting of 119881(119866) = 119883 and

119864(119866) = (119899(119899+1) (119899+1)(119899+2)) 119899 isin Ncup(1199092119899 1199092119899+1

)

119899 isin N 119909 isin [0 1] cup ((11990922119899

) 0) 119899 isin N 119909 isin [0 1] Note that119866 is not a (119862)-graph as 119899(119899 + 1) rarr 1 Define 119891 119883 rarr 119883 as119891119909 = 1199092 Then 119866 is a (119862

119891)-graph since 119891

119899119909 = 1199092

119899rarr 0

4 Abstract and Applied Analysis

Example 16 Let 119883 = 1119899 119899 isin N cup 0 cup N with respect to119887-metric 119889(119909 119910) = |119909 minus 119910|

2 and let 119868 be identity map on 119883Consider a graph 119866

2consisting of 119881(119866

2) = 119883 and

119864 (1198662) = (

1

119899

1

119899 + 1) (

1

119899 + 1 119899) (119899 0)

(1

5119899 0) 119899 isin N

(12)

since 119909119899

= 1119899 rarr 0 as 119899 rarr infin We note that 1198662is a (119862

119868)-

graph but 119903(119909119899 0) = 2|(1119899) minus (1(119899 + 1))|

2+ 22|(1(119899 + 1)) minus

119899|2+ 221198992999424999426999456 0 as 119899 rarr infin Thus 119866

2is not an (119867

119868)-graph

Example 17 Let119883 = 1119899 119899 isin Ncupradic5119899 119899 isin Ncup0withrespect to 119887-metric119889(119909 119910) = |119909minus119910|

2 and let 119868 be identitymapon 119883 Consider a graph 119866

3consisting of 119881(119866

3) = 119883 and

119864 (1198663) = (

1

119899

1

119899 + 1) (

1

119899 + 1radic5

119899) (

radic5

119899 0) 119899 isin N

(13)

since 119909119899

= 1119899 rarr 0 as 119899 rarr infin Clearly 1198663is not a (119862

119868)-

graph but it is easy to verify that 1198663is an (119867

119868)-graph

Above examples show that for a given 119891 notions (119862119891)-

graph and (119867119891)-graph are independent even if 119891 is identity

map

Theorem 18 Let (119883 119889) be a complete 119887-metric space and let119891 be a 119887-(120593 119866) contraction where 120593 is 119887-comparison functionAssume that 119889 is continuous and there is 119911

0in 119883 for which

(1199110 1198911199110) is an edge in 119866 Then the following assertions hold

(1) If 119866 is a (119862119891)-graph then 119891 has a unique fixed point

119905 isin [1199110]119866 and for any 119910 isin [119911

0]119866 119891119899119910 rarr 119905 Further if

119866 is weakly connected then 119891 is Picard operator

(2) If 119866 is weakly connected (119867119891)-graph then 119891 has a

unique fixed point 119905 isin 119883 and for any 119910 isin 119883 119891119899119910 rarr 119905

Proof (1) It follows from Proposition 13 that 1198911198991199110 is a

Cauchy sequence in119883 Since119883 is complete there exists 119905 isin 119883

such that 1198911198991199110

rarr 119905 Since (1198911198991199110 119891119899+1

1199110) isin 119864(119866) for all

119899 isin N and 119866 is a (119862119891) graphs there exists a subsequence

1198911198991198961199110 of 119891

1198991199110 and 119901 isin N such that (119891

1198991198961199110 119905) isin 119864(119866) for

all 119896 ge 119901 Observe that (1199110 1198911199110 11989121199110 119891

11989911199110 119891

1198991199011199110 119905)

is a path in 119866 Therefore 119905 isin [1199110]119866 From (2) we get

119889 (119891119899119896+11199110 119891119905) le 120593 (119889 (119891

1198991198961199110 119905)) lt 119889 (119891

1198991198961199110 119905) forall119896 ge 119899

0

(14)

Letting 119896 rarr infin we obtain lim119896rarrinfin

119891119899119896+11199110

= 119891119905 as 119889

is continuous Since 1198911198991198961199110 is a subsequence of 119891

1198991199110 we

conclude that 119891119905 = 119905 Finally if 119910 isin [1199110]119866 it follows from

Lemma 12 that 119891119899119910 rarr 119905(2) Let 119866 be weakly connected (119867

119891)-graph From

Proposition 13 1198911198991199110

rarr 119905 isin 119883 and then 119903(1198911198991199110 119905) rarr 0 as

119899 rarr infin Now for each 119899 isin N let 119910119899

119894 119894 = 0 1 119872

119899be a

path from 1198911198991199110to 119905 with 119910

0= 119905 and 119910

119899

119872119899

= 1198911198991199110in 119866 then

119889 (119905 119891119905) le 119904 [119889 (119905 119891119899+1

1199110) + 119889 (119891

119899+1

1199110 119891119905)]

le 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

119889 (119891119910119899

119894minus1 119891119910119899

119894)]

le 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

120593 (119889 (119910119899

119894minus1 119910119899

119894))]

lt 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

119889 (119910119899

119894minus1 119910119899

119894)]

= 119904 [119889 (119905 119891119899+1

1199110) + 119903 (119891

119899

1199110 119905)]

(15)

Letting 119899 rarr infin the above inequality yields 119891119905 = 119905 Let 119910 isin

[1199110]119866

= 119883 be arbitrary then from Lemma 12 and Remark 10it is easily seen that 119891119899119910 rarr 119905

The following example shows that the condition of (119862119891)

or (119867119891)-graph in the hypothesis of Theorem 18 can not be

dropped

Example 19 Let119883 = [0 1] let 119889(119909 119910) = |119909minus119910|2 and let119891119909 =

1199092 for all 119909 isin (0 1] and 1198910 = 12 Then (119883 119889) is a complete119887-metric space with 119904 = 2 Further 119889 is continuous and 119891 isa 119887-(120593 119866

1) contraction (with 120593(119905) = 1199054) where 119881(119866

1) = 119883

and 119864(1198661) = (119909 119910) isin (0 1] times (0 1] 119909 ge 119910 cup (0 0) (0 1)

Note that 1198661is weakly connected but 119891 has no fixed point in

[1199110]1198661

= 119883 Observe that 1198661is not a (119862

119891)-graph because the

sequence 119891119899119909 = 1199092

119899rarr 0 for 119909 isin (0 1] and (119891

119899119909 119891119899+1

119909) isin

119864(1198661) 119899 isin N but it does not contain any subsequence such

that (119909119899119896

0) isin 119864(1198661) Also we note that for any fixed 119909 isin

(0 1] 119903(119891119899119909 0) = 2[|1199092119899minus 1|2

+ |1 minus 0|2

] 999424999426999456 0 as 119899 rarr infin

Definition 20 Let (119883 119889) be a 119887-metric space A mapping 119891

119883 rarr 119883 is called orbitally continuous if for all 119909 119910 isin 119883 andany sequence 119896

119899119899isinN of positive integers 119891119896119899119909 rarr 119910 implies

119891(119891119896119899119909) rarr 119891119910 as 119899 rarr infin A mapping 119891 119883 rarr 119883 is called

orbitally 119866-continuous if for all 119909 119910 isin 119883 and any sequence119896119899119899isinN of positive integers 119891119896119899119909 rarr 119910 and (119891

119896119899119909 119891119896119899+1119909) isin

119864(119866) for all 119899 isin N imply 119891(119891119896119899119909) rarr 119891119910

Theorem21 Let (119883 119889) be a complete 119887-metric space and let119891be a 119887-(120593 119866) contraction where 120593 is a 119887-comparison functionAssume that 119889 is continuous 119891 is orbitally 119866-continuous andthere is 119911

0in119883 for which (119911

0 1198911199110) is an edge in 119866 Then 119891 has

a fixed point 119905 isin 119883 Moreover for any 119910 isin [1199110]119866 119891119899119910 rarr 119905

Proof It follows from Proposition 13 that 1198911198991199110 is a Cauchy

sequence in (119883 119889) Since 119883 is complete there exists 119905 isin 119883

such that lim119899rarrinfin

1198911198991199110

= 119905 Since (1198911198991199110 119891119899+1

1199110) isin 119864(119866) for

all 119899 isin N 119891 is orbitally 119866-continuous Therefore continuityof 119889 implies that 119891119905 = 119905 Let 119910 isin [119911

0]119866be arbitrary then it

follows from Lemma 12 that lim119899rarrinfin

119891119899119910 = 119905

Abstract and Applied Analysis 5

Slightly strengthening the continuity condition on 119891 ournext theorem deals with the graph 119866 which may fail to havethe property that there is 119911

0in119883 for which (119911

0 1198911199110) is an edge

in 119866

Theorem22 Let (119883 119889) be a complete 119887-metric space and let119891be a 119887-(120593 119866) contraction where 120593 is a 119887-comparison functionAssume that 119889 is continuous 119891 is orbitally continuous andthere is 119911

0in119883 for which 119891119911

0isin [1199110]119866 Then for any 119910 isin [119911

0]119866

119891119899119910 rarr 119905 isin 119883 where 119905 is a fixed point of 119891

Proof It follows from Proposition 13 that 1198911198991199110 is a Cauchy

sequence in 119883 Since 119883 is complete there exists 119905 isin 119883

such that 1198911198991199110

rarr 119905 Since 119891 is orbitally continuous thenlim119899rarrinfin

1198911198911198991199110

= 119891119905 which yields 119891119905 = 119905 Let 119910 isin [1199110]119866be

arbitrary then from Lemma 12 lim119899rarrinfin

119891119899119910 = 119905

Remark 23 In addition to the hypothesis ofTheorems 21 and22 if we assume that 119866 is weakly connected then 119891 willbecome Picard operator [11] on 119883

Remark 24 Theorem 18 generalizesextends claims 40 and

50 of [10 Theorem 32] and [7 Theorem 4(1)] Theorem 21generalizes claims 20 and 3

0 of [10Theorem 33]Theorem 22generalizes claims 2

0 and 30 of [10 Theorem 34] and thus

generalizes extends results ofNieto andRodrıguez-Lopez [24Theorems 21 and 23] Petrusel andRus [11Theorem43] andRan and Reurings [25 Theorem 21] We mention here thatTheorem 18 can not be improved using comparison functioninstead of 119887-comparison function (see Gwozdz-Łukawskaand Jachymski [26 Example 2])

We observe that Theorem 22 can be used to extendfamous fixed point theorem of Edelstein to the case of119887-metric space We need to define notion of 120598-chainableproperty for 119887-metric space

Definition 25 A 119887-metric space (119883 119889) is said to be 120598-chainable for some 120598 gt 0 if for119909 119910 isin 119883 there exist119909

119894isin 119883 119894 =

0 1 2 119897 with 1199090

= 119909 119909119897= 119910 such that 119889(119909

119894minus1 119909119894) lt 120598 for

119894 = 1 2 119897

Corollary 26 Let (119883 119889) be a complete 120598-chainable 119887-metricspace Assume that 119889 is continuous and there exists a 119887-comparison function 120593 R+ rarr R+ such that 119891 119883 rarr 119883

satisfying

119889 (119909 119910) lt 120598 119894119898119901119897119894119890119904 119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) (16)

for all 119909 119910 isin 119883 Then 119891 is a Picard operator

Proof Consider a graph 119866 consisting of 119881(119866) = 119883 and(119909 119910) isin 119864(119866) if and only if 119889(119909 119910) lt 120598 Since119883 is 120598-chainable119866 is weakly connected Let (119909 119910) isin 119864(119866) and from (16) wehave

119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) lt 119889 (119909 119910) lt 120598 (17)

Then 119889(119891119909 119891119910) isin 119864(119866) Therefore in view of (16) 119891 is 119887-(120593 119866) contraction Further (16) implies that 119891 is continuousNow the conclusion follows by usingTheorem 22

Now we establish a fixed point theorem using a generalcontractive condition

Theorem 27 Let (119883 119889) be a complete 119887-metric space and let119866 be a (119862

119891)-graph in119883times119883 such that119881(119866) = 119883 and 119891 119883 rarr

119883 is an edge-preserving mapping Assume that 119889 is continuousand there exist 120572 120573 120574 ge 0 with 119904120572 + (119904 + 1)120573 + 119904(119904 + 1)120574 lt 1

and for all (119909 119910) isin 119864(119866)

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(18)

If there is 1199110in119883 for which (119911

0 1198911199110) is an edge in119866 then119891 has

a fixed point in [1199110]119866

Proof Since 119891 is edge-preserving then (1198911198991199110 119891119899+1

1199110) isin

119864(119866) for all 119899 isin N From (18) and using (d3) it follows that

119889 (119891119899

1199110 119891119899+1

1199110)

le 120572119889 (119891119899minus1

1199110 119891119899

1199110)

+ 120573 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1z0)]

+ 120574119904 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1

1199110)]

(19)

On rearranging

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904] 119889 (119891

119899minus1

1199110 119891119899

1199110) (20)

Repeating iteratively we have

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119899

119889 (1199110 1198911199110) (21)

For 119898 gt 119899 ge 1 and using (d3) Definition 1 we have

119889 (119891119899

1199110 119891119898

1199110)

le 119904119889 (119891119899

1199110 119891119899+1

1199110) + 1199042

119889 (119891119899+1

1199110 119891119899+2

1199110)

+ sdot sdot sdot + 119904119898minus119899

119889 (119891119898minus1

1199110 119891119898

1199110)

le1

119904119899minus1119889 (1199110 1198911199110)

119898minus1

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(using (21))

lt1

119904119899minus1119889 (1199110 1198911199110)

infin

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(22)

Since 119904[(120572 + 120573 + 120574119904)(1 minus 120573 minus 120574119904)] lt 1 then 1198911198991199110 is a

Cauchy sequence in 119883 By completeness of 119883 the sequence1198911198991199110 converges to 119905 isin 119883 Since 119866 is a (119862

119891)-graph there

exists a subsequence 1198911198991198961199110 and anatural number119901 such that

6 Abstract and Applied Analysis

(1198911198991198961199110 119905) isin 119864(119866) for all 119896 ge 119901 Using (18) for all 119896 ge 119901 we

have

119889 (119891119899119896+11199110 119891119905)

le 120572119889 (1198911198991198961199110 119905) + 120573 [119889 (119891

1198991198961199110 119891119899119896+11199110) + 119889 (119905 119891119905)]

+ 120574 [119889 (1198911198991198961199110 119891119905) + 119889 (119905 119891

119899119896+11199110)]

(23)

Since the 119887-metric 119889 is continuous and 120573 + 120574 lt 1 letting119896 rarr infin inequality (23) yields 119891119905 = 119905 Also note that(1199110 1198911199110 11989121199110 119891

11989911199110 119891

119899119901 119905) is a path in 119866 and hencein 119866 therefore 119905 isin [119911

0]119866

We note thatTheorem 27 does not guarantee the unique-ness of fixed point but this can be accomplished under someassumptions as in the following theorem

Theorem 28 In addition to the hypothesis ofTheorem 27 onefurther assumes that if 120572 + 2119904120573 + 2120574 lt 1 for the same set of120572 120573 120574 ge 0 and for any two fixed points 119905

1 1199052exist119911 isin 119883 such that

(1199051 119911) and (119905

2 119911) isin 119864(119866) Then 119891 has a unique fixed point

Proof Let 1199051and 1199052be two fixed points of 119891 and then there

exists 119911 isin 119883 such that (1199051 119911) (119905

2 119911) isin 119864(119866) By induction we

have (1199051 119891119899119911) (1199052 119891119899119911) isin 119864(119866) for all 119899 = 0 1 From (18)

we have

119889 (1199051 119891119899

119911) le 120572119889 (1199051 119891119899minus1

119911) + 120573119889 (119891119899minus1

119911 119891119899

119911)

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

le 120572119889 (1199051 119891119899minus1

119911) + 120573119904 [119889 (119891119899minus1

119911 1199051) + 119889 (119905

1 119891119899

119911)]

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

(24)

On rearranging

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574] 119889 (1199051 119891119899minus1

119911) forall119899 = 1 2

(25)

Continuing recursively (25) gives

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574]

119899

119889 (1199051 119911) (26)

Since [(120572+119904120573+120574)(1minus119904120573minus120574)] lt 1 then lim119899rarrinfin

119889(1199051 119891119899119911) =

0 Similarly one can show that lim119899rarrinfin

119889(1199052 119891119899119911) = 0 which

by using (d3) Definition 1 infers that 119889(1199051 1199052) = 0

Suppose that (119883 ⪯) is a partially ordered set Considergraph 119866

2consisting of 119864(119866

2) = (119909 119910) isin 119883 times 119883 119909 ⪯

119910 or119910 ⪯ 119909 and 119881(1198662) coincides with 119883 We note that

if a self-mapping 119891 is monotone with respect to the orderthen for graph 119866

2 it is obvious that 119891 is edge-preserving

or equivalently we can say that 119891 maps comparable elementsonto comparable elements

Following corollaries are the direct consequences ofTheorem 28

Corollary 29 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists 120572 120573 120574 ge 0 with 120572 + 2120573 + 2120574 lt 1 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(27)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 30 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (28)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 31 Let (119883 119889) be a complete metric space where 119883

is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (29)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Remark 32 Wenote that inTheorem 27 the condition ldquothereis 1199110in 119883 for which (119911

0 1198911199110) is an edge in 119866rdquo yields 119891

1198991199110

rarr

119905 where 119905 isin 119883 is a fixed point of 119891 Consider a graph119866 = (119883119883 times 119883) For such graph under the assumptions ofTheorems 27 and 28 it infers that119891 is a Picard operatorThusmany standard fixed point theorems can be easily deducedfromTheorem 28 as follows

Abstract and Applied Analysis 7

Corollary 33 (Hardy and Rogers [27]) Let (119883 119889) be acomplete metric space and let 119891 119883 rarr 119883 Suppose that thereexists constants 120572 120573 120574 ge 0 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(30)

for all 119909 119910 isin 119883 where 120572 + 2120573 + 2120574 lt 1 then 119891 has a uniquefixed point in 119883

Corollary 34 (Kannan [28]) Let (119883 119889) be a complete metricspace and let119891 119883 rarr 119883 Suppose that there exists a constants119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (31)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

Corollary 35 (Chatterjea [29]) Let (119883 119889) be a completemetric space and let 119891 119883 rarr 119883 Suppose that there existsa constants 119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (32)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

3 Applications

Let 119883 be a nonempty set let 119898 be a positive integer 119883119894119898

119894=1

be nonempty closed subsets of 119883 and let 119891 ⋃119898

119894=1119883119894

rarr

⋃119898

119894=1119883119894be an operatorThen119883 = ⋃

119898

119894=1119883119894is known as cyclic

representation of 119883 wrt 119891 if

119891 (1198831) sub 119883

2 119891 (119883

119898minus1) sub 119883

119898 119891 (119883

119898) sub 119883

1

(33)

and operator 119891 is known as cyclic operator [30]

Theorem 36 Let (119883 119889) be complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

120593 R+ rarr R+ be a 119887-comparison function and let 119891 119884 rarr

119884 Further suppose that(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) 119889(119891119909 119891119910) le 120593(119889(119909 119910)) whenever 119909 isin 119883119894 119910 isin 119883

119894+1

where 119883119898+1

= 1198831

Then 119891 has a unique fixed point 119905 isin ⋂119898

119894=1119883119894and 119891

119899119910 rarr 119905 for

any 119910 isin ⋃119898

119894=1119883119894

Proof We note that (119884 119889) is complete 119887-metric space Let usconsider a graph 119866 consisting of 119881(119866) = 119884 and 119864(119866) =

Δ cup (119909 119910) isin 119884 times 119884 119909 isin 119883119894 119910 isin 119883

119894+1 119894 = 1 119898 By

(i) it follows that 119891 preserves edges Now let 119891119899119909 rarr 119909

lowast in119884 such that (119891119899119909 119891

119899+1119909) isin 119864(119866) for all 119899 ge 1 then in view of

(33) sequence 119891119899119909 has infinitely many terms in each 119883

119894so

that one can easily extract a subsequence of 119891119899119909 convergingto 119909lowast in each 119883

119894 Since 119883

119894rsquos are closed then 119909

lowastisin ⋂119898

119894=1119883119894

Now it is easy to form a subsequence 119891119899119896119909 in some 119883

119895

119895 isin 1 119898 such that (119891119899119896119909 119909lowast) isin 119864(119866) for 119896 ge 1 and

it indicates that 119866 is weakly connected (119862119891)-graph and thus

conclusion follows fromTheorem 18

Remark 37 (see [31Theorem 21(1)]) It can be deduced fromTheorem 36 if (119883 119889) is a metric space

On the same lines as in proof of Theorem 36 we obtainthe following consequence of Theorem 28

Theorem38 Let (119883 119889) be a complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

and let 119891 119884 rarr 119884 Further suppose that

(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) there exist 120572 120573 120574 ge 0with 119904120572+119904(119904+1)120573+119904(119904+1)120574 lt 1

such that

119889 (119891119909 119891119910)

le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(34)

whenever 119909 isin 119883119894 119910 isin 119883

119894+1 where 119883

119898+1= 1198831

Then 119891 has a fixed point 119905 isin ⋂119898

119894=1119883119894

Remark 39 [32 Theorem 7] and [33 Theorem 31] can bededuced from Theorem 38 but it does not ensure 119891 to be aPicard operator

Remark 40 We note that in proof [32 Theorem 7] theauthorrsquos argument to prove that is 119866 (as assumed in proofof Theorem 36) a (119862)-graph is valid only if the terms ofsequence 119909

119899 are Picard iterations otherwise it is void For

example let 119884 = ⋃3

119894=1119883119894where 119883

1= 1119899 119899 is odd cup 0

1198832= 1119899 119899 is even cup 0 119883

3= 0 and define 119891 119884 rarr 119884

as 119891119909 = 1199092 119909 isin 119884 1198832and 119891119909 = 0 119909 isin 119883

2 We see that (33)

is satisfied and 1119899 rarr 0 but 1198833does not contain infinitely

many terms of 119909119899 In the following we give the corrected

argument to prove that 119866 is a (119862)-graph

Let 119909119899

rarr 119909 in 119883 such that (119909119899 119909119899+1

) isin 119864(119866) for all119899 ge 1 Keeping in mind construction of119866 there exists at leastone pair of closed sets 119883

119895 119883119895+1

for some 119895 isin 1 2 119898

such that both sets contain infinitely many terms of sequence119909119899 since 119883

119894rsquos are closed so that 119909 isin 119883

119895cap 119883119895+1

for some119895 isin 1 119898 and thus one can easily extract a subsequencesuch that (119909

119899119896 119909) isin 119864(119866) holds for 119896 ge 1

Now we establish an existence theorem for the solutionof an integral equation as a consequence of our Theorem 18

Theorem 41 Consider the integral equation

119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (35)

where 119896 [119886 119887] times [119886 119887] times R119899 rarr R119899 and 119892 [119886 119887] rarr R119899 iscontinuous Assume that

(i) 119896(119905 119904 sdot) R119899 rarr R119899 is nondecreasing for each 119905 119904 isin

[119886 119887](ii) there exists a (119888)-comparison function 120593 R+ rarr R+

and a continuous function 119901 [119886 119887] times [119886 119887] rarr R+

8 Abstract and Applied Analysis

such that |119896(119905 119904 119909(119904)) minus 119896(119905 119904 119910(119904))| le 119901(119905 119904)120593(|119909(119904) minus

119910(119904)|) for each 119905 119904 isin [119886 119887] and 119909 le 119910 (ie 119909(119905) le 119910(119905)

for all 119905 isin [119886 119887])

(iii) sup119905isin[119886119887]

int119887

119886119901(119905 119904)119889119904 le 1

(iv) there exists 1199090

isin 119862([119886 119887]R119899) such that 1199090(119905) le

int119887

119886119896(119905 119904 119909

0(119904))119889119904 + 119892(119905) for all 119905 isin [119886 119887]

Then the integral equation (35) has a unique solution in theset 119909 isin 119862([119886 119887]R119899) 119909(119905) le 119909

0(119905) or 119909(119905) ge 119909

0(119905) for all

119905 isin [119886 119887]

Proof Let (119883 sdot infin

) where 119883 = 119862([119886 119887]R119899) and define amapping 119879 119862([119886 119887]R119899) rarr 119862([119886 119887]R119899) by

119879119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (36)

Consider a graph 119866 consisting of 119881(119866) = 119883 and 119864(119866) =

(119909 119910) isin 119883 times 119883 119909(119905) le 119910(119905) for all 119905 isin [119886 119887] From property(i) we observe that the mapping 119879 is nondecreasing thus 119879

preserves edges Furthermore 119866 is a (119862)-graph that is forevery nondecreasing sequence 119909

119899 sub 119883 which converges to

some 119911 isin 119883 then 119909119899(119905) le 119911(119905) for all 119905 isin [119886 119887] Now for every

119909 119910 isin 119883 with (119909 119910) isin 119864(119866) we have

1003816100381610038161003816119879119909 (119905) minus 119879119910 (119905)1003816100381610038161003816

le int

119887

119886

1003816100381610038161003816119896 (119905 119904 119909 (119904)) minus 119896 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904

le int

119887

119886

119901 (119905 119904) 120593 (1003816100381610038161003816119909 (119904) minus 119910 (119904)

1003816100381610038161003816) 119889119904

le 120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817infin) int

119887

119886

119901 (119905 119904) 119889119904

(37)

Hence 119889(119879119909 119879119910) le 120593(119889(119909 119910)) From (iv) we have(1199090 1198791199090) isin 119864(119866) so that [119909

0]119866

= 119909 isin 119862([119886 119887]R119899) 119909(119905) le

1199090(119905) or 119909(119905) ge 119909

0(119905) for all 119905 isin [119886 119887] The conclusion follows

fromTheorem 18

Note that Theorem 41 specifies region of solution whichinvokes the novelty of our result

References

[1] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacesrdquo Journal of Functional Analysis vol 30 pp 26ndash37 1989

[2] J Heinonen Lectures on Analysis on Metric Spaces SpringerNew York NY USA 2001

[3] N Bourbaki Topologie Generale Herman Paris France 1974[4] S Czerwik ldquoNonlinear set-valued contraction mappings in

119887-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[5] M Bota A Molnar and C Varga ldquoOn Ekelandrsquos variationalprinciple in 119887-metric spacesrdquo Fixed Point Theory vol 12 no 1pp 21ndash28 2011

[6] M F Bota-Boriceanu andA Petrusel ldquoUlam-Hyers stability foroperatorial equationsrdquo Analele Stiintifice ale Universitatii ldquoAl ICuzardquo din Iasi vol 57 supplement 1 pp 65ndash74 2011

[7] M Pacurar ldquoA fixed point result for 120593-contractions on 119887-metricspaces without the boundedness assumptionrdquo PolytechnicaPosnaniensis InstitutumMathematicum Fasciculi Mathematicino 43 pp 127ndash137 2010

[8] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis Babes-Bolyai vol 41 no 4pp 23ndash27 1996

[9] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[10] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[11] A Petrusel and I A Rus ldquoFixed point theorems in ordered l-spacesrdquo Proceedings of the American Mathematical Society vol134 no 2 pp 411ndash418 2006

[12] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 p 68 1975

[13] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[14] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008

[15] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[16] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997

[17] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquoSeminar on Fixed Point Theory vol 3 pp 3ndash9 1993

[18] T P Petru and M Boriceanu ldquoFixed point results for gener-alized 120593-contraction on a set with two metricsrdquo TopologicalMethods in Nonlinear Analysis vol 33 no 2 pp 315ndash326 2009

[19] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 11 pages 2012

[20] A Amini-Harandi ldquoCoupled and tripled fixed point theory inpartially ordered metric spaces with application to initial valueproblemrdquoMathematical and Computer Modelling vol 57 no 9-10 pp 2343ndash2348 2012

[21] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[22] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012

[23] M Samreen and T Kamran ldquoFixed point theorems for integralG-contractionsrdquo Fixed Point Theory and Applications vol 2013article 149 2013

[24] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[25] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrix

Abstract and Applied Analysis 9

equationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[26] G Gwozdz-Łukawska and J Jachymski ldquoIFS on a metric spacewith a graph structure and extensions of the Kelisky-Rivlintheoremrdquo Journal of Mathematical Analysis and Applicationsvol 356 no 2 pp 453ndash463 2009

[27] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973

[28] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968

[29] S K Chatterjea ldquoFixed-point theoremsrdquo Doklady BolgarskoıAkademii Nauk vol 25 pp 727ndash730 1972

[30] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003

[31] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010

[32] F Bojor ldquoFixed point theorems for Reich type contractionson metric spaces with a graphrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 9 pp 3895ndash3901 2012

[33] M A Petric ldquoSome remarks concerning Ciric-Reich-Rus oper-atorsrdquo Creative Mathematics and Informatics vol 18 no 2 pp188ndash193 2009

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

metric fixed point theory and subsequently further studywas developed in this setting by different authors whenunderlying space was taken to be a partially ordered set (seeeg [13 14])

Let 120593 R+ rarr R+ Consider the following properties

(i)120593

1199051le 1199052rArr 120593(119905

1) le 120593(119905

2) for all 119905

1 1199052isin R+

(ii)120593

120593(119905) lt 119905 for 119905 gt 0

(iii)120593

120593(0) = 0

(iv)120593lim119899rarrinfin

120593119899(119905) = 0 for all 119905 ge 0

(v)120593

suminfin

119899=0120593119899(119905) converges for all 119905 gt 0

It is easily seen that (i)120593and (iv)

120593imply (ii)

120593and (i)

120593and (ii)

120593

imply (iii)120593

We recall that a function 120593 satisfying (i)120593and (iv)

120593is said

to be a comparison function A function 120593 satisfying (i)120593and

(v)120593is known as (119888)-comparison functionAny (119888)-comparison function is a comparison function

but converse may not be true For example 120593(119905) = 119905(1 + 119905)119905 isin R+ is a comparison function but not a (119888)-comparisonfunction On the other hand define 120593(119905) = 1199052 0 le 119905 le 1

and 120593(119905) = 119905 minus (12) 119905 gt 1 and then 120593 is a (119888)-comparisonfunction For details on 120593 contractions we refer the readersto [15 16]Berinde [17] took further step to investigate 120593 contractionswhen the framework was taken to be a 119887-metric space andfor some technical reasons he had to introduce the notionof 119887-comparison function in particular he obtained someestimations for rate of convergence [17] For other relatedresults see also [5 7 17ndash21]

Definition 3 Let 119904 ge 1 be a fixed real number A function 120593

R+ rarr R+ is known as 119887-comparison function if it satisfies(i)120593 and the following holds

(vi)120593

suminfin

119899=0119904119899120593119899(119905) converges for all 119905 isin R+

The concept of 119887-comparison function coincides withcomparison function when 119904 = 1 Let (119883 119889) be a 119887-metricspace with coefficient 119904 ge 1 and then 120593(119905) = 119886119905 119905 isin R+ with0 lt 119886 lt (1119904) is a 119887-comparison function

2 Main Results

Throughout this section let (119883 119889) be a 119887-metric space withcoefficient 119904 ge 1 and Δ is the diagonal of the Cartesianproduct 119883 times 119883 119866 is a directed graph such that the set 119881(119866)

of its vertices coincides with 119883 and the set 119864(119866) of its edgescontains all loops that is 119864(119866) supe Δ Assume that 119866 has noparallel edges We assign to each edge having vertices 119909 and119910 a unique element 119889(119909 119910) Now we introduce the followingdefinition

Definition 4 One says that a mapping 119891 119883 rarr 119883 is a 119887-(120593 119866) contraction if for all 119909 119910 isin 119883

(119891119909 119891119910) isin 119864 (119866) whenever (119909 119910) isin 119864 (119866) (1)

119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) whenever (119909 119910) isin 119864 (119866)

(2)

where 120593 R+ rarr R+ is a comparison function

Remark 5 Note that a Banach 119866-contraction is a 119887-(120593 119866)

contraction

Example 6 Any constant mapping 119891 119883 rarr 119883 is a 119887-(120593 119866)

contraction for any graph 119866 with 119881(119866) = 119883

Example 7 Any self-mapping 119891 on 119883 is trivially a 119887-(120593 1198661)

contraction where 1198661= (119881(119866) 119864(119866)) = (119883 Δ)

Example 8 Let 119883 = R and define 119889 119883 times 119883 rarr R by119889(119909 119910) = |119909 minus 119910|

2 Then 119889 is a 119887-metric on 119883 with 119904 = 2Further 119891119909 = 1199092 for all 119909 isin 119883 Then 119891 is a 119887-(120593 119866

0)

contraction with 120593(119905) = 1199054 and 1198660= (119883119883times119883) Note that 119889

is not a metric on 119883

Definition 9 Two sequences 119909119899 and 119910

119899 in119883 are said to be

equivalent if lim119899rarrinfin

119889(119909119899 119910119899) = 0 and if each of them is a

Cauchy sequence then they are called Cauchy equivalent

As a consequence of Definition 9 we get the followinglemma

Remark 10 Let 119909119899 and 119910

119899 be equivalent sequences in 119883

(i) If 119909119899 converges to 119909 then 119910

119899 also converges to 119909 and

vice versa (ii) 119910119899 is a Cauchy sequence whenever 119909

119899 is a

Cauchy sequence and vice versa

Now we recollect some preliminaries from graph theorywhich we need for the sequel Let 119866 = (119881(119866) and let 119864(119866))

be a directed graph By letter 119866 we denote the undirectedgraph obtained from 119866 by ignoring the direction of edgesIf 119909 and 119910 are vertices in a graph 119866 then a path in 119866 from119909 to 119910 of length 119897 is a sequence 119909

119894119897

119894=0of 119897 + 1 vertices such

that 1199090

= 119909 119909119897

= 119910 and (119909119894minus1

119909119894) isin 119864(119866) for 119894 = 1 119897

A graph 119866 is called connected if there is a path between anytwo vertices 119866 is weakly connected if 119866 is connected For agraph 119866 such that 119864(119866) is symmetric and 119909 is a vertex in 119866the subgraph 119866

119909consisting of all edges and vertices which

are contained in some path beginning at 119909 is known as acomponent of 119866 containing 119909 So that 119881(119866

119909) = [119909]

119866 where

[119909]119866is the equivalence class of a relation 119877 defined on 119881(119866)

by the rule 119910119877119911 if there is a path in119866 from 119910 to 119911 Clearly119866119909

is connected A graph119866 is known as (119862)-graph in119883 [22] if forany sequence 119909

119899 in 119883 with 119909

119899rarr 119909 and (119909

119899 119909119899+1

) isin 119864(119866)

for 119899 isin N then there exists a subsequence 119909119899119896

of 119909119899 such

that (119909119899119896

119909) isin 119864(119866) for 119896 isin N

Proposition 11 Let 119891 119883 rarr 119883 be a 119887-(120593 119866) contractionwhere 120593 R+ rarr R+ is a comparison function then

(i) 119891 is a 119887-(120593 119866) contraction and a 119887-(120593 119866minus1

) contractionas well

(ii) [1199090]119866is 119891-invariant and 119891|

[1199090]119866

is a 119887-(120593 119866119909) contrac-

tion provided that 1199090isin 119883 is such that 119891119909

0isin [1199090]119866

Abstract and Applied Analysis 3

Proof (i) By using (d2) (Definition 1) it can be easily proved(ii) Let 119909 isin [119909

0]119866Then there is a path 119909 = 119911

0 1199111 119911

119897=

1199090between 119909 and 119909

0 Since 119891 is a 119887-(120593 119866) contraction then

(119891119911119894minus1

119891119911119894) isin 119864(119866) for all 119894 = 1 2 119897 Thus 119891119909 isin [119891119909

0]119866

=

[1199090]119866

Suppose (119909 119910) isin 119864(1198661199090

) Then (119891119909 119891119910) isin 119864(119866) since119891 is a 119887-(120593 119866) contraction But [119909

0]119866is 119891 invariant so we

conclude that (119891119909 119891119910) isin 119864(1198661199090

) Condition (2) is satisfiedautomatically as 119866

1199090is a subgraph of 119866

Fromnowon we assume that coefficient of 119887-comparisonfunction is at least as large as the coefficient of 119887-metric 119904

Lemma 12 Let 119891 119883 rarr 119883 be a 119887-(120593 119866) contraction where120593 R+ rarr R+ is a 119887-comparison function Then given any119909 isin 119883 and 119910 isin [119909]

119866 two sequences 119891

119899119909 and 119891

119899119910 are

equivalent

Proof Let 119909 isin 119883 and let 119910 isin [119909]119866 then there exists a path

119909119894119897

119894=0in 119866 from 119909 to 119910 with 119909

0= 119909 119909

119897= 119910 and (119909

119894minus1 119909119894) isin

119864(119866) From Proposition 11 119891 is a 119887-(120593 119866) contraction So

(119891119899

119909119894minus1

119891119899

119909119894) isin 119864 (119866) implies 119889 (119891

119899

119909119894minus1

119891119899

119909119894)

le 120593 (119889 (119891119899minus1

119909119894minus1

119891119899minus1

119909119894))

(3)

for all 119899 isin N and 119894 = 0 1 2 119897 Hence

119889 (119891119899

119909119894minus1

119891119899

119909119894) le 120593119899

(119889 (119909119894minus1

119909119894))

forall119899 isin N 119894 = 0 1 2 119897

(4)

We observe that 119891119899119909119894119897

119894=0is a path in 119866 from 119891

119899119909 to 119891

119899119910

Using (d3) Definition 1 and (4) we have

119889 (119891119899

119909 119891119899

119910) le

119897

sum

119894=1

119904119894

119889 (119891119899

119909119894minus1

119891119899

119909119894)

le

119897

sum

119894=1

119904119894

120593119899

(119889 (119909119894minus1

119909119894))

(5)

Letting 119899 rarr infin we obtain 119889(119891119899119909 119891119899119910) rarr 0

Proposition 13 Let 119891 be a 119887-(120593 119866) contraction where 120593

R+ rarr R+ is a 119887-comparison function Suppose that there is 1199110

in 119883 such that 1198911199110isin [1199110]119866 Then 119891119899119911

0 is a Cauchy sequence

in 119883

Proof Since 1198911199110

isin [1199110]119866 let 119910

119894119903

119894=0be a path from 119911

0to 1198911199110

Then using the same arguments as in Lemma 12 we arrive at

119889 (119891119899

1199110 119891119899+1

1199110) le

119903

sum

119894=1

119904119894

120593119899

(119889 (119910119894minus1

119910119894)) forall119899 isin N

(6)

Let 119898 gt 119899 ge 1 and then from above inequality it follows for119901 ge 1

119889 (119891119899

1199110 119891119899+119901

1199110)

le 119904119889 (119891119899

1199110 119891119899+1

1199110) + 1199042

119889 (119891119899+1

1199110 119891119899+2

1199110)

+ sdot sdot sdot + 119904119901

119889 (119891119899+119901minus1

1199110 119891119899+119901

1199110)

le1

119904119899minus1[

[

119899+119901minus1

sum

119895=119899

119904119895

119889 (119891119895

1199110 119891119895+1

1199110)]

]

le1

119904119899minus1[

[

119903

sum

119894=1

119904119894

119899+119901minus1

sum

119895=119899

119904119895

120593119895

(119889 (119910119894minus1

119910119894))]

]

(7)

Denoting for each 119894 = 1 2 119903

119878119894

119899=

119899

sum

119896=0

119904119896

120593119896

(119889 (119910119894minus1

119910119894)) 119899 ge 1 (8)

relation (7) becomes

119889 (119891119899

1199110 119891119899+119901

1199110) le

1

119904119899minus1[

119903

sum

119894=1

119904119894

[119878119894

119899+119901minus1minus 119878119894

119899minus1]] (9)

since 120593 is a 119887-comparison function so that for each 119894 =

1 2 119903infin

sum

119896=0

119904119896

120593119896

(119889 (119910119894minus1

119910119894)) lt infin (10)

Then corresponding to each 119894 there is a real number 119878119894 such

that

lim119899rarrinfin

119878119894

119899= 119878119894

(11)

In view of (11) relation (9) gives 119889(1198911198991199110 119891119899+119901

1199110) rarr 0 as

119899 rarr infin Which shows that 1198911198991199110 is a Cauchy sequence in

119883

Definition 14 Let 119891 119883 rarr 119883 and let 119910 isin 119883 andthe sequence 119891

119899119910 in 119883 is such that 119891

119899119910 rarr 119909

lowast with(119891119899119910 119891119899+1

119910) isin 119864(119866) for 119899 isin N One says that a graph 119866 is(119862119891)-graph if there exists a subsequence 119891

119899119896119910 and a naturalnumber 119901 such that (119891119899119896119910 119909

lowast) isin 119864(119866) for all 119896 ge 119901 [23] One

says that a graph 119866 is (119867119891)-graph if 119891

119899119910 isin [119909

lowast]119866for 119899 ge 1

then 119903(119891119899119910 119909lowast) rarr 0 (as 119899 rarr infin) where 119903(119891

119899119910 119909lowast) =

sum119872

119894=1119904119894119889(119911119894minus1

119911119894) 119911119894119872

119894=0is a path from 119891

119899119910 to 119909

lowast in 119866

Obviously every (119862)-graph is a (119862119891)-graph for any self-

mapping 119891 on 119883 but converse may not hold as shown in thefollowing

Example 15 Let119883 = [0 1] with respect to 119887-metric 119889(119909 119910) =

|119909 minus 119910|2 Consider a graph 119866 consisting of 119881(119866) = 119883 and

119864(119866) = (119899(119899+1) (119899+1)(119899+2)) 119899 isin Ncup(1199092119899 1199092119899+1

)

119899 isin N 119909 isin [0 1] cup ((11990922119899

) 0) 119899 isin N 119909 isin [0 1] Note that119866 is not a (119862)-graph as 119899(119899 + 1) rarr 1 Define 119891 119883 rarr 119883 as119891119909 = 1199092 Then 119866 is a (119862

119891)-graph since 119891

119899119909 = 1199092

119899rarr 0

4 Abstract and Applied Analysis

Example 16 Let 119883 = 1119899 119899 isin N cup 0 cup N with respect to119887-metric 119889(119909 119910) = |119909 minus 119910|

2 and let 119868 be identity map on 119883Consider a graph 119866

2consisting of 119881(119866

2) = 119883 and

119864 (1198662) = (

1

119899

1

119899 + 1) (

1

119899 + 1 119899) (119899 0)

(1

5119899 0) 119899 isin N

(12)

since 119909119899

= 1119899 rarr 0 as 119899 rarr infin We note that 1198662is a (119862

119868)-

graph but 119903(119909119899 0) = 2|(1119899) minus (1(119899 + 1))|

2+ 22|(1(119899 + 1)) minus

119899|2+ 221198992999424999426999456 0 as 119899 rarr infin Thus 119866

2is not an (119867

119868)-graph

Example 17 Let119883 = 1119899 119899 isin Ncupradic5119899 119899 isin Ncup0withrespect to 119887-metric119889(119909 119910) = |119909minus119910|

2 and let 119868 be identitymapon 119883 Consider a graph 119866

3consisting of 119881(119866

3) = 119883 and

119864 (1198663) = (

1

119899

1

119899 + 1) (

1

119899 + 1radic5

119899) (

radic5

119899 0) 119899 isin N

(13)

since 119909119899

= 1119899 rarr 0 as 119899 rarr infin Clearly 1198663is not a (119862

119868)-

graph but it is easy to verify that 1198663is an (119867

119868)-graph

Above examples show that for a given 119891 notions (119862119891)-

graph and (119867119891)-graph are independent even if 119891 is identity

map

Theorem 18 Let (119883 119889) be a complete 119887-metric space and let119891 be a 119887-(120593 119866) contraction where 120593 is 119887-comparison functionAssume that 119889 is continuous and there is 119911

0in 119883 for which

(1199110 1198911199110) is an edge in 119866 Then the following assertions hold

(1) If 119866 is a (119862119891)-graph then 119891 has a unique fixed point

119905 isin [1199110]119866 and for any 119910 isin [119911

0]119866 119891119899119910 rarr 119905 Further if

119866 is weakly connected then 119891 is Picard operator

(2) If 119866 is weakly connected (119867119891)-graph then 119891 has a

unique fixed point 119905 isin 119883 and for any 119910 isin 119883 119891119899119910 rarr 119905

Proof (1) It follows from Proposition 13 that 1198911198991199110 is a

Cauchy sequence in119883 Since119883 is complete there exists 119905 isin 119883

such that 1198911198991199110

rarr 119905 Since (1198911198991199110 119891119899+1

1199110) isin 119864(119866) for all

119899 isin N and 119866 is a (119862119891) graphs there exists a subsequence

1198911198991198961199110 of 119891

1198991199110 and 119901 isin N such that (119891

1198991198961199110 119905) isin 119864(119866) for

all 119896 ge 119901 Observe that (1199110 1198911199110 11989121199110 119891

11989911199110 119891

1198991199011199110 119905)

is a path in 119866 Therefore 119905 isin [1199110]119866 From (2) we get

119889 (119891119899119896+11199110 119891119905) le 120593 (119889 (119891

1198991198961199110 119905)) lt 119889 (119891

1198991198961199110 119905) forall119896 ge 119899

0

(14)

Letting 119896 rarr infin we obtain lim119896rarrinfin

119891119899119896+11199110

= 119891119905 as 119889

is continuous Since 1198911198991198961199110 is a subsequence of 119891

1198991199110 we

conclude that 119891119905 = 119905 Finally if 119910 isin [1199110]119866 it follows from

Lemma 12 that 119891119899119910 rarr 119905(2) Let 119866 be weakly connected (119867

119891)-graph From

Proposition 13 1198911198991199110

rarr 119905 isin 119883 and then 119903(1198911198991199110 119905) rarr 0 as

119899 rarr infin Now for each 119899 isin N let 119910119899

119894 119894 = 0 1 119872

119899be a

path from 1198911198991199110to 119905 with 119910

0= 119905 and 119910

119899

119872119899

= 1198911198991199110in 119866 then

119889 (119905 119891119905) le 119904 [119889 (119905 119891119899+1

1199110) + 119889 (119891

119899+1

1199110 119891119905)]

le 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

119889 (119891119910119899

119894minus1 119891119910119899

119894)]

le 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

120593 (119889 (119910119899

119894minus1 119910119899

119894))]

lt 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

119889 (119910119899

119894minus1 119910119899

119894)]

= 119904 [119889 (119905 119891119899+1

1199110) + 119903 (119891

119899

1199110 119905)]

(15)

Letting 119899 rarr infin the above inequality yields 119891119905 = 119905 Let 119910 isin

[1199110]119866

= 119883 be arbitrary then from Lemma 12 and Remark 10it is easily seen that 119891119899119910 rarr 119905

The following example shows that the condition of (119862119891)

or (119867119891)-graph in the hypothesis of Theorem 18 can not be

dropped

Example 19 Let119883 = [0 1] let 119889(119909 119910) = |119909minus119910|2 and let119891119909 =

1199092 for all 119909 isin (0 1] and 1198910 = 12 Then (119883 119889) is a complete119887-metric space with 119904 = 2 Further 119889 is continuous and 119891 isa 119887-(120593 119866

1) contraction (with 120593(119905) = 1199054) where 119881(119866

1) = 119883

and 119864(1198661) = (119909 119910) isin (0 1] times (0 1] 119909 ge 119910 cup (0 0) (0 1)

Note that 1198661is weakly connected but 119891 has no fixed point in

[1199110]1198661

= 119883 Observe that 1198661is not a (119862

119891)-graph because the

sequence 119891119899119909 = 1199092

119899rarr 0 for 119909 isin (0 1] and (119891

119899119909 119891119899+1

119909) isin

119864(1198661) 119899 isin N but it does not contain any subsequence such

that (119909119899119896

0) isin 119864(1198661) Also we note that for any fixed 119909 isin

(0 1] 119903(119891119899119909 0) = 2[|1199092119899minus 1|2

+ |1 minus 0|2

] 999424999426999456 0 as 119899 rarr infin

Definition 20 Let (119883 119889) be a 119887-metric space A mapping 119891

119883 rarr 119883 is called orbitally continuous if for all 119909 119910 isin 119883 andany sequence 119896

119899119899isinN of positive integers 119891119896119899119909 rarr 119910 implies

119891(119891119896119899119909) rarr 119891119910 as 119899 rarr infin A mapping 119891 119883 rarr 119883 is called

orbitally 119866-continuous if for all 119909 119910 isin 119883 and any sequence119896119899119899isinN of positive integers 119891119896119899119909 rarr 119910 and (119891

119896119899119909 119891119896119899+1119909) isin

119864(119866) for all 119899 isin N imply 119891(119891119896119899119909) rarr 119891119910

Theorem21 Let (119883 119889) be a complete 119887-metric space and let119891be a 119887-(120593 119866) contraction where 120593 is a 119887-comparison functionAssume that 119889 is continuous 119891 is orbitally 119866-continuous andthere is 119911

0in119883 for which (119911

0 1198911199110) is an edge in 119866 Then 119891 has

a fixed point 119905 isin 119883 Moreover for any 119910 isin [1199110]119866 119891119899119910 rarr 119905

Proof It follows from Proposition 13 that 1198911198991199110 is a Cauchy

sequence in (119883 119889) Since 119883 is complete there exists 119905 isin 119883

such that lim119899rarrinfin

1198911198991199110

= 119905 Since (1198911198991199110 119891119899+1

1199110) isin 119864(119866) for

all 119899 isin N 119891 is orbitally 119866-continuous Therefore continuityof 119889 implies that 119891119905 = 119905 Let 119910 isin [119911

0]119866be arbitrary then it

follows from Lemma 12 that lim119899rarrinfin

119891119899119910 = 119905

Abstract and Applied Analysis 5

Slightly strengthening the continuity condition on 119891 ournext theorem deals with the graph 119866 which may fail to havethe property that there is 119911

0in119883 for which (119911

0 1198911199110) is an edge

in 119866

Theorem22 Let (119883 119889) be a complete 119887-metric space and let119891be a 119887-(120593 119866) contraction where 120593 is a 119887-comparison functionAssume that 119889 is continuous 119891 is orbitally continuous andthere is 119911

0in119883 for which 119891119911

0isin [1199110]119866 Then for any 119910 isin [119911

0]119866

119891119899119910 rarr 119905 isin 119883 where 119905 is a fixed point of 119891

Proof It follows from Proposition 13 that 1198911198991199110 is a Cauchy

sequence in 119883 Since 119883 is complete there exists 119905 isin 119883

such that 1198911198991199110

rarr 119905 Since 119891 is orbitally continuous thenlim119899rarrinfin

1198911198911198991199110

= 119891119905 which yields 119891119905 = 119905 Let 119910 isin [1199110]119866be

arbitrary then from Lemma 12 lim119899rarrinfin

119891119899119910 = 119905

Remark 23 In addition to the hypothesis ofTheorems 21 and22 if we assume that 119866 is weakly connected then 119891 willbecome Picard operator [11] on 119883

Remark 24 Theorem 18 generalizesextends claims 40 and

50 of [10 Theorem 32] and [7 Theorem 4(1)] Theorem 21generalizes claims 20 and 3

0 of [10Theorem 33]Theorem 22generalizes claims 2

0 and 30 of [10 Theorem 34] and thus

generalizes extends results ofNieto andRodrıguez-Lopez [24Theorems 21 and 23] Petrusel andRus [11Theorem43] andRan and Reurings [25 Theorem 21] We mention here thatTheorem 18 can not be improved using comparison functioninstead of 119887-comparison function (see Gwozdz-Łukawskaand Jachymski [26 Example 2])

We observe that Theorem 22 can be used to extendfamous fixed point theorem of Edelstein to the case of119887-metric space We need to define notion of 120598-chainableproperty for 119887-metric space

Definition 25 A 119887-metric space (119883 119889) is said to be 120598-chainable for some 120598 gt 0 if for119909 119910 isin 119883 there exist119909

119894isin 119883 119894 =

0 1 2 119897 with 1199090

= 119909 119909119897= 119910 such that 119889(119909

119894minus1 119909119894) lt 120598 for

119894 = 1 2 119897

Corollary 26 Let (119883 119889) be a complete 120598-chainable 119887-metricspace Assume that 119889 is continuous and there exists a 119887-comparison function 120593 R+ rarr R+ such that 119891 119883 rarr 119883

satisfying

119889 (119909 119910) lt 120598 119894119898119901119897119894119890119904 119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) (16)

for all 119909 119910 isin 119883 Then 119891 is a Picard operator

Proof Consider a graph 119866 consisting of 119881(119866) = 119883 and(119909 119910) isin 119864(119866) if and only if 119889(119909 119910) lt 120598 Since119883 is 120598-chainable119866 is weakly connected Let (119909 119910) isin 119864(119866) and from (16) wehave

119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) lt 119889 (119909 119910) lt 120598 (17)

Then 119889(119891119909 119891119910) isin 119864(119866) Therefore in view of (16) 119891 is 119887-(120593 119866) contraction Further (16) implies that 119891 is continuousNow the conclusion follows by usingTheorem 22

Now we establish a fixed point theorem using a generalcontractive condition

Theorem 27 Let (119883 119889) be a complete 119887-metric space and let119866 be a (119862

119891)-graph in119883times119883 such that119881(119866) = 119883 and 119891 119883 rarr

119883 is an edge-preserving mapping Assume that 119889 is continuousand there exist 120572 120573 120574 ge 0 with 119904120572 + (119904 + 1)120573 + 119904(119904 + 1)120574 lt 1

and for all (119909 119910) isin 119864(119866)

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(18)

If there is 1199110in119883 for which (119911

0 1198911199110) is an edge in119866 then119891 has

a fixed point in [1199110]119866

Proof Since 119891 is edge-preserving then (1198911198991199110 119891119899+1

1199110) isin

119864(119866) for all 119899 isin N From (18) and using (d3) it follows that

119889 (119891119899

1199110 119891119899+1

1199110)

le 120572119889 (119891119899minus1

1199110 119891119899

1199110)

+ 120573 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1z0)]

+ 120574119904 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1

1199110)]

(19)

On rearranging

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904] 119889 (119891

119899minus1

1199110 119891119899

1199110) (20)

Repeating iteratively we have

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119899

119889 (1199110 1198911199110) (21)

For 119898 gt 119899 ge 1 and using (d3) Definition 1 we have

119889 (119891119899

1199110 119891119898

1199110)

le 119904119889 (119891119899

1199110 119891119899+1

1199110) + 1199042

119889 (119891119899+1

1199110 119891119899+2

1199110)

+ sdot sdot sdot + 119904119898minus119899

119889 (119891119898minus1

1199110 119891119898

1199110)

le1

119904119899minus1119889 (1199110 1198911199110)

119898minus1

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(using (21))

lt1

119904119899minus1119889 (1199110 1198911199110)

infin

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(22)

Since 119904[(120572 + 120573 + 120574119904)(1 minus 120573 minus 120574119904)] lt 1 then 1198911198991199110 is a

Cauchy sequence in 119883 By completeness of 119883 the sequence1198911198991199110 converges to 119905 isin 119883 Since 119866 is a (119862

119891)-graph there

exists a subsequence 1198911198991198961199110 and anatural number119901 such that

6 Abstract and Applied Analysis

(1198911198991198961199110 119905) isin 119864(119866) for all 119896 ge 119901 Using (18) for all 119896 ge 119901 we

have

119889 (119891119899119896+11199110 119891119905)

le 120572119889 (1198911198991198961199110 119905) + 120573 [119889 (119891

1198991198961199110 119891119899119896+11199110) + 119889 (119905 119891119905)]

+ 120574 [119889 (1198911198991198961199110 119891119905) + 119889 (119905 119891

119899119896+11199110)]

(23)

Since the 119887-metric 119889 is continuous and 120573 + 120574 lt 1 letting119896 rarr infin inequality (23) yields 119891119905 = 119905 Also note that(1199110 1198911199110 11989121199110 119891

11989911199110 119891

119899119901 119905) is a path in 119866 and hencein 119866 therefore 119905 isin [119911

0]119866

We note thatTheorem 27 does not guarantee the unique-ness of fixed point but this can be accomplished under someassumptions as in the following theorem

Theorem 28 In addition to the hypothesis ofTheorem 27 onefurther assumes that if 120572 + 2119904120573 + 2120574 lt 1 for the same set of120572 120573 120574 ge 0 and for any two fixed points 119905

1 1199052exist119911 isin 119883 such that

(1199051 119911) and (119905

2 119911) isin 119864(119866) Then 119891 has a unique fixed point

Proof Let 1199051and 1199052be two fixed points of 119891 and then there

exists 119911 isin 119883 such that (1199051 119911) (119905

2 119911) isin 119864(119866) By induction we

have (1199051 119891119899119911) (1199052 119891119899119911) isin 119864(119866) for all 119899 = 0 1 From (18)

we have

119889 (1199051 119891119899

119911) le 120572119889 (1199051 119891119899minus1

119911) + 120573119889 (119891119899minus1

119911 119891119899

119911)

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

le 120572119889 (1199051 119891119899minus1

119911) + 120573119904 [119889 (119891119899minus1

119911 1199051) + 119889 (119905

1 119891119899

119911)]

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

(24)

On rearranging

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574] 119889 (1199051 119891119899minus1

119911) forall119899 = 1 2

(25)

Continuing recursively (25) gives

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574]

119899

119889 (1199051 119911) (26)

Since [(120572+119904120573+120574)(1minus119904120573minus120574)] lt 1 then lim119899rarrinfin

119889(1199051 119891119899119911) =

0 Similarly one can show that lim119899rarrinfin

119889(1199052 119891119899119911) = 0 which

by using (d3) Definition 1 infers that 119889(1199051 1199052) = 0

Suppose that (119883 ⪯) is a partially ordered set Considergraph 119866

2consisting of 119864(119866

2) = (119909 119910) isin 119883 times 119883 119909 ⪯

119910 or119910 ⪯ 119909 and 119881(1198662) coincides with 119883 We note that

if a self-mapping 119891 is monotone with respect to the orderthen for graph 119866

2 it is obvious that 119891 is edge-preserving

or equivalently we can say that 119891 maps comparable elementsonto comparable elements

Following corollaries are the direct consequences ofTheorem 28

Corollary 29 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists 120572 120573 120574 ge 0 with 120572 + 2120573 + 2120574 lt 1 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(27)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 30 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (28)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 31 Let (119883 119889) be a complete metric space where 119883

is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (29)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Remark 32 Wenote that inTheorem 27 the condition ldquothereis 1199110in 119883 for which (119911

0 1198911199110) is an edge in 119866rdquo yields 119891

1198991199110

rarr

119905 where 119905 isin 119883 is a fixed point of 119891 Consider a graph119866 = (119883119883 times 119883) For such graph under the assumptions ofTheorems 27 and 28 it infers that119891 is a Picard operatorThusmany standard fixed point theorems can be easily deducedfromTheorem 28 as follows

Abstract and Applied Analysis 7

Corollary 33 (Hardy and Rogers [27]) Let (119883 119889) be acomplete metric space and let 119891 119883 rarr 119883 Suppose that thereexists constants 120572 120573 120574 ge 0 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(30)

for all 119909 119910 isin 119883 where 120572 + 2120573 + 2120574 lt 1 then 119891 has a uniquefixed point in 119883

Corollary 34 (Kannan [28]) Let (119883 119889) be a complete metricspace and let119891 119883 rarr 119883 Suppose that there exists a constants119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (31)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

Corollary 35 (Chatterjea [29]) Let (119883 119889) be a completemetric space and let 119891 119883 rarr 119883 Suppose that there existsa constants 119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (32)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

3 Applications

Let 119883 be a nonempty set let 119898 be a positive integer 119883119894119898

119894=1

be nonempty closed subsets of 119883 and let 119891 ⋃119898

119894=1119883119894

rarr

⋃119898

119894=1119883119894be an operatorThen119883 = ⋃

119898

119894=1119883119894is known as cyclic

representation of 119883 wrt 119891 if

119891 (1198831) sub 119883

2 119891 (119883

119898minus1) sub 119883

119898 119891 (119883

119898) sub 119883

1

(33)

and operator 119891 is known as cyclic operator [30]

Theorem 36 Let (119883 119889) be complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

120593 R+ rarr R+ be a 119887-comparison function and let 119891 119884 rarr

119884 Further suppose that(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) 119889(119891119909 119891119910) le 120593(119889(119909 119910)) whenever 119909 isin 119883119894 119910 isin 119883

119894+1

where 119883119898+1

= 1198831

Then 119891 has a unique fixed point 119905 isin ⋂119898

119894=1119883119894and 119891

119899119910 rarr 119905 for

any 119910 isin ⋃119898

119894=1119883119894

Proof We note that (119884 119889) is complete 119887-metric space Let usconsider a graph 119866 consisting of 119881(119866) = 119884 and 119864(119866) =

Δ cup (119909 119910) isin 119884 times 119884 119909 isin 119883119894 119910 isin 119883

119894+1 119894 = 1 119898 By

(i) it follows that 119891 preserves edges Now let 119891119899119909 rarr 119909

lowast in119884 such that (119891119899119909 119891

119899+1119909) isin 119864(119866) for all 119899 ge 1 then in view of

(33) sequence 119891119899119909 has infinitely many terms in each 119883

119894so

that one can easily extract a subsequence of 119891119899119909 convergingto 119909lowast in each 119883

119894 Since 119883

119894rsquos are closed then 119909

lowastisin ⋂119898

119894=1119883119894

Now it is easy to form a subsequence 119891119899119896119909 in some 119883

119895

119895 isin 1 119898 such that (119891119899119896119909 119909lowast) isin 119864(119866) for 119896 ge 1 and

it indicates that 119866 is weakly connected (119862119891)-graph and thus

conclusion follows fromTheorem 18

Remark 37 (see [31Theorem 21(1)]) It can be deduced fromTheorem 36 if (119883 119889) is a metric space

On the same lines as in proof of Theorem 36 we obtainthe following consequence of Theorem 28

Theorem38 Let (119883 119889) be a complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

and let 119891 119884 rarr 119884 Further suppose that

(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) there exist 120572 120573 120574 ge 0with 119904120572+119904(119904+1)120573+119904(119904+1)120574 lt 1

such that

119889 (119891119909 119891119910)

le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(34)

whenever 119909 isin 119883119894 119910 isin 119883

119894+1 where 119883

119898+1= 1198831

Then 119891 has a fixed point 119905 isin ⋂119898

119894=1119883119894

Remark 39 [32 Theorem 7] and [33 Theorem 31] can bededuced from Theorem 38 but it does not ensure 119891 to be aPicard operator

Remark 40 We note that in proof [32 Theorem 7] theauthorrsquos argument to prove that is 119866 (as assumed in proofof Theorem 36) a (119862)-graph is valid only if the terms ofsequence 119909

119899 are Picard iterations otherwise it is void For

example let 119884 = ⋃3

119894=1119883119894where 119883

1= 1119899 119899 is odd cup 0

1198832= 1119899 119899 is even cup 0 119883

3= 0 and define 119891 119884 rarr 119884

as 119891119909 = 1199092 119909 isin 119884 1198832and 119891119909 = 0 119909 isin 119883

2 We see that (33)

is satisfied and 1119899 rarr 0 but 1198833does not contain infinitely

many terms of 119909119899 In the following we give the corrected

argument to prove that 119866 is a (119862)-graph

Let 119909119899

rarr 119909 in 119883 such that (119909119899 119909119899+1

) isin 119864(119866) for all119899 ge 1 Keeping in mind construction of119866 there exists at leastone pair of closed sets 119883

119895 119883119895+1

for some 119895 isin 1 2 119898

such that both sets contain infinitely many terms of sequence119909119899 since 119883

119894rsquos are closed so that 119909 isin 119883

119895cap 119883119895+1

for some119895 isin 1 119898 and thus one can easily extract a subsequencesuch that (119909

119899119896 119909) isin 119864(119866) holds for 119896 ge 1

Now we establish an existence theorem for the solutionof an integral equation as a consequence of our Theorem 18

Theorem 41 Consider the integral equation

119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (35)

where 119896 [119886 119887] times [119886 119887] times R119899 rarr R119899 and 119892 [119886 119887] rarr R119899 iscontinuous Assume that

(i) 119896(119905 119904 sdot) R119899 rarr R119899 is nondecreasing for each 119905 119904 isin

[119886 119887](ii) there exists a (119888)-comparison function 120593 R+ rarr R+

and a continuous function 119901 [119886 119887] times [119886 119887] rarr R+

8 Abstract and Applied Analysis

such that |119896(119905 119904 119909(119904)) minus 119896(119905 119904 119910(119904))| le 119901(119905 119904)120593(|119909(119904) minus

119910(119904)|) for each 119905 119904 isin [119886 119887] and 119909 le 119910 (ie 119909(119905) le 119910(119905)

for all 119905 isin [119886 119887])

(iii) sup119905isin[119886119887]

int119887

119886119901(119905 119904)119889119904 le 1

(iv) there exists 1199090

isin 119862([119886 119887]R119899) such that 1199090(119905) le

int119887

119886119896(119905 119904 119909

0(119904))119889119904 + 119892(119905) for all 119905 isin [119886 119887]

Then the integral equation (35) has a unique solution in theset 119909 isin 119862([119886 119887]R119899) 119909(119905) le 119909

0(119905) or 119909(119905) ge 119909

0(119905) for all

119905 isin [119886 119887]

Proof Let (119883 sdot infin

) where 119883 = 119862([119886 119887]R119899) and define amapping 119879 119862([119886 119887]R119899) rarr 119862([119886 119887]R119899) by

119879119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (36)

Consider a graph 119866 consisting of 119881(119866) = 119883 and 119864(119866) =

(119909 119910) isin 119883 times 119883 119909(119905) le 119910(119905) for all 119905 isin [119886 119887] From property(i) we observe that the mapping 119879 is nondecreasing thus 119879

preserves edges Furthermore 119866 is a (119862)-graph that is forevery nondecreasing sequence 119909

119899 sub 119883 which converges to

some 119911 isin 119883 then 119909119899(119905) le 119911(119905) for all 119905 isin [119886 119887] Now for every

119909 119910 isin 119883 with (119909 119910) isin 119864(119866) we have

1003816100381610038161003816119879119909 (119905) minus 119879119910 (119905)1003816100381610038161003816

le int

119887

119886

1003816100381610038161003816119896 (119905 119904 119909 (119904)) minus 119896 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904

le int

119887

119886

119901 (119905 119904) 120593 (1003816100381610038161003816119909 (119904) minus 119910 (119904)

1003816100381610038161003816) 119889119904

le 120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817infin) int

119887

119886

119901 (119905 119904) 119889119904

(37)

Hence 119889(119879119909 119879119910) le 120593(119889(119909 119910)) From (iv) we have(1199090 1198791199090) isin 119864(119866) so that [119909

0]119866

= 119909 isin 119862([119886 119887]R119899) 119909(119905) le

1199090(119905) or 119909(119905) ge 119909

0(119905) for all 119905 isin [119886 119887] The conclusion follows

fromTheorem 18

Note that Theorem 41 specifies region of solution whichinvokes the novelty of our result

References

[1] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacesrdquo Journal of Functional Analysis vol 30 pp 26ndash37 1989

[2] J Heinonen Lectures on Analysis on Metric Spaces SpringerNew York NY USA 2001

[3] N Bourbaki Topologie Generale Herman Paris France 1974[4] S Czerwik ldquoNonlinear set-valued contraction mappings in

119887-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[5] M Bota A Molnar and C Varga ldquoOn Ekelandrsquos variationalprinciple in 119887-metric spacesrdquo Fixed Point Theory vol 12 no 1pp 21ndash28 2011

[6] M F Bota-Boriceanu andA Petrusel ldquoUlam-Hyers stability foroperatorial equationsrdquo Analele Stiintifice ale Universitatii ldquoAl ICuzardquo din Iasi vol 57 supplement 1 pp 65ndash74 2011

[7] M Pacurar ldquoA fixed point result for 120593-contractions on 119887-metricspaces without the boundedness assumptionrdquo PolytechnicaPosnaniensis InstitutumMathematicum Fasciculi Mathematicino 43 pp 127ndash137 2010

[8] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis Babes-Bolyai vol 41 no 4pp 23ndash27 1996

[9] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[10] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[11] A Petrusel and I A Rus ldquoFixed point theorems in ordered l-spacesrdquo Proceedings of the American Mathematical Society vol134 no 2 pp 411ndash418 2006

[12] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 p 68 1975

[13] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[14] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008

[15] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[16] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997

[17] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquoSeminar on Fixed Point Theory vol 3 pp 3ndash9 1993

[18] T P Petru and M Boriceanu ldquoFixed point results for gener-alized 120593-contraction on a set with two metricsrdquo TopologicalMethods in Nonlinear Analysis vol 33 no 2 pp 315ndash326 2009

[19] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 11 pages 2012

[20] A Amini-Harandi ldquoCoupled and tripled fixed point theory inpartially ordered metric spaces with application to initial valueproblemrdquoMathematical and Computer Modelling vol 57 no 9-10 pp 2343ndash2348 2012

[21] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[22] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012

[23] M Samreen and T Kamran ldquoFixed point theorems for integralG-contractionsrdquo Fixed Point Theory and Applications vol 2013article 149 2013

[24] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[25] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrix

Abstract and Applied Analysis 9

equationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[26] G Gwozdz-Łukawska and J Jachymski ldquoIFS on a metric spacewith a graph structure and extensions of the Kelisky-Rivlintheoremrdquo Journal of Mathematical Analysis and Applicationsvol 356 no 2 pp 453ndash463 2009

[27] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973

[28] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968

[29] S K Chatterjea ldquoFixed-point theoremsrdquo Doklady BolgarskoıAkademii Nauk vol 25 pp 727ndash730 1972

[30] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003

[31] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010

[32] F Bojor ldquoFixed point theorems for Reich type contractionson metric spaces with a graphrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 9 pp 3895ndash3901 2012

[33] M A Petric ldquoSome remarks concerning Ciric-Reich-Rus oper-atorsrdquo Creative Mathematics and Informatics vol 18 no 2 pp188ndash193 2009

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

Abstract and Applied Analysis 3

Proof (i) By using (d2) (Definition 1) it can be easily proved(ii) Let 119909 isin [119909

0]119866Then there is a path 119909 = 119911

0 1199111 119911

119897=

1199090between 119909 and 119909

0 Since 119891 is a 119887-(120593 119866) contraction then

(119891119911119894minus1

119891119911119894) isin 119864(119866) for all 119894 = 1 2 119897 Thus 119891119909 isin [119891119909

0]119866

=

[1199090]119866

Suppose (119909 119910) isin 119864(1198661199090

) Then (119891119909 119891119910) isin 119864(119866) since119891 is a 119887-(120593 119866) contraction But [119909

0]119866is 119891 invariant so we

conclude that (119891119909 119891119910) isin 119864(1198661199090

) Condition (2) is satisfiedautomatically as 119866

1199090is a subgraph of 119866

Fromnowon we assume that coefficient of 119887-comparisonfunction is at least as large as the coefficient of 119887-metric 119904

Lemma 12 Let 119891 119883 rarr 119883 be a 119887-(120593 119866) contraction where120593 R+ rarr R+ is a 119887-comparison function Then given any119909 isin 119883 and 119910 isin [119909]

119866 two sequences 119891

119899119909 and 119891

119899119910 are

equivalent

Proof Let 119909 isin 119883 and let 119910 isin [119909]119866 then there exists a path

119909119894119897

119894=0in 119866 from 119909 to 119910 with 119909

0= 119909 119909

119897= 119910 and (119909

119894minus1 119909119894) isin

119864(119866) From Proposition 11 119891 is a 119887-(120593 119866) contraction So

(119891119899

119909119894minus1

119891119899

119909119894) isin 119864 (119866) implies 119889 (119891

119899

119909119894minus1

119891119899

119909119894)

le 120593 (119889 (119891119899minus1

119909119894minus1

119891119899minus1

119909119894))

(3)

for all 119899 isin N and 119894 = 0 1 2 119897 Hence

119889 (119891119899

119909119894minus1

119891119899

119909119894) le 120593119899

(119889 (119909119894minus1

119909119894))

forall119899 isin N 119894 = 0 1 2 119897

(4)

We observe that 119891119899119909119894119897

119894=0is a path in 119866 from 119891

119899119909 to 119891

119899119910

Using (d3) Definition 1 and (4) we have

119889 (119891119899

119909 119891119899

119910) le

119897

sum

119894=1

119904119894

119889 (119891119899

119909119894minus1

119891119899

119909119894)

le

119897

sum

119894=1

119904119894

120593119899

(119889 (119909119894minus1

119909119894))

(5)

Letting 119899 rarr infin we obtain 119889(119891119899119909 119891119899119910) rarr 0

Proposition 13 Let 119891 be a 119887-(120593 119866) contraction where 120593

R+ rarr R+ is a 119887-comparison function Suppose that there is 1199110

in 119883 such that 1198911199110isin [1199110]119866 Then 119891119899119911

0 is a Cauchy sequence

in 119883

Proof Since 1198911199110

isin [1199110]119866 let 119910

119894119903

119894=0be a path from 119911

0to 1198911199110

Then using the same arguments as in Lemma 12 we arrive at

119889 (119891119899

1199110 119891119899+1

1199110) le

119903

sum

119894=1

119904119894

120593119899

(119889 (119910119894minus1

119910119894)) forall119899 isin N

(6)

Let 119898 gt 119899 ge 1 and then from above inequality it follows for119901 ge 1

119889 (119891119899

1199110 119891119899+119901

1199110)

le 119904119889 (119891119899

1199110 119891119899+1

1199110) + 1199042

119889 (119891119899+1

1199110 119891119899+2

1199110)

+ sdot sdot sdot + 119904119901

119889 (119891119899+119901minus1

1199110 119891119899+119901

1199110)

le1

119904119899minus1[

[

119899+119901minus1

sum

119895=119899

119904119895

119889 (119891119895

1199110 119891119895+1

1199110)]

]

le1

119904119899minus1[

[

119903

sum

119894=1

119904119894

119899+119901minus1

sum

119895=119899

119904119895

120593119895

(119889 (119910119894minus1

119910119894))]

]

(7)

Denoting for each 119894 = 1 2 119903

119878119894

119899=

119899

sum

119896=0

119904119896

120593119896

(119889 (119910119894minus1

119910119894)) 119899 ge 1 (8)

relation (7) becomes

119889 (119891119899

1199110 119891119899+119901

1199110) le

1

119904119899minus1[

119903

sum

119894=1

119904119894

[119878119894

119899+119901minus1minus 119878119894

119899minus1]] (9)

since 120593 is a 119887-comparison function so that for each 119894 =

1 2 119903infin

sum

119896=0

119904119896

120593119896

(119889 (119910119894minus1

119910119894)) lt infin (10)

Then corresponding to each 119894 there is a real number 119878119894 such

that

lim119899rarrinfin

119878119894

119899= 119878119894

(11)

In view of (11) relation (9) gives 119889(1198911198991199110 119891119899+119901

1199110) rarr 0 as

119899 rarr infin Which shows that 1198911198991199110 is a Cauchy sequence in

119883

Definition 14 Let 119891 119883 rarr 119883 and let 119910 isin 119883 andthe sequence 119891

119899119910 in 119883 is such that 119891

119899119910 rarr 119909

lowast with(119891119899119910 119891119899+1

119910) isin 119864(119866) for 119899 isin N One says that a graph 119866 is(119862119891)-graph if there exists a subsequence 119891

119899119896119910 and a naturalnumber 119901 such that (119891119899119896119910 119909

lowast) isin 119864(119866) for all 119896 ge 119901 [23] One

says that a graph 119866 is (119867119891)-graph if 119891

119899119910 isin [119909

lowast]119866for 119899 ge 1

then 119903(119891119899119910 119909lowast) rarr 0 (as 119899 rarr infin) where 119903(119891

119899119910 119909lowast) =

sum119872

119894=1119904119894119889(119911119894minus1

119911119894) 119911119894119872

119894=0is a path from 119891

119899119910 to 119909

lowast in 119866

Obviously every (119862)-graph is a (119862119891)-graph for any self-

mapping 119891 on 119883 but converse may not hold as shown in thefollowing

Example 15 Let119883 = [0 1] with respect to 119887-metric 119889(119909 119910) =

|119909 minus 119910|2 Consider a graph 119866 consisting of 119881(119866) = 119883 and

119864(119866) = (119899(119899+1) (119899+1)(119899+2)) 119899 isin Ncup(1199092119899 1199092119899+1

)

119899 isin N 119909 isin [0 1] cup ((11990922119899

) 0) 119899 isin N 119909 isin [0 1] Note that119866 is not a (119862)-graph as 119899(119899 + 1) rarr 1 Define 119891 119883 rarr 119883 as119891119909 = 1199092 Then 119866 is a (119862

119891)-graph since 119891

119899119909 = 1199092

119899rarr 0

4 Abstract and Applied Analysis

Example 16 Let 119883 = 1119899 119899 isin N cup 0 cup N with respect to119887-metric 119889(119909 119910) = |119909 minus 119910|

2 and let 119868 be identity map on 119883Consider a graph 119866

2consisting of 119881(119866

2) = 119883 and

119864 (1198662) = (

1

119899

1

119899 + 1) (

1

119899 + 1 119899) (119899 0)

(1

5119899 0) 119899 isin N

(12)

since 119909119899

= 1119899 rarr 0 as 119899 rarr infin We note that 1198662is a (119862

119868)-

graph but 119903(119909119899 0) = 2|(1119899) minus (1(119899 + 1))|

2+ 22|(1(119899 + 1)) minus

119899|2+ 221198992999424999426999456 0 as 119899 rarr infin Thus 119866

2is not an (119867

119868)-graph

Example 17 Let119883 = 1119899 119899 isin Ncupradic5119899 119899 isin Ncup0withrespect to 119887-metric119889(119909 119910) = |119909minus119910|

2 and let 119868 be identitymapon 119883 Consider a graph 119866

3consisting of 119881(119866

3) = 119883 and

119864 (1198663) = (

1

119899

1

119899 + 1) (

1

119899 + 1radic5

119899) (

radic5

119899 0) 119899 isin N

(13)

since 119909119899

= 1119899 rarr 0 as 119899 rarr infin Clearly 1198663is not a (119862

119868)-

graph but it is easy to verify that 1198663is an (119867

119868)-graph

Above examples show that for a given 119891 notions (119862119891)-

graph and (119867119891)-graph are independent even if 119891 is identity

map

Theorem 18 Let (119883 119889) be a complete 119887-metric space and let119891 be a 119887-(120593 119866) contraction where 120593 is 119887-comparison functionAssume that 119889 is continuous and there is 119911

0in 119883 for which

(1199110 1198911199110) is an edge in 119866 Then the following assertions hold

(1) If 119866 is a (119862119891)-graph then 119891 has a unique fixed point

119905 isin [1199110]119866 and for any 119910 isin [119911

0]119866 119891119899119910 rarr 119905 Further if

119866 is weakly connected then 119891 is Picard operator

(2) If 119866 is weakly connected (119867119891)-graph then 119891 has a

unique fixed point 119905 isin 119883 and for any 119910 isin 119883 119891119899119910 rarr 119905

Proof (1) It follows from Proposition 13 that 1198911198991199110 is a

Cauchy sequence in119883 Since119883 is complete there exists 119905 isin 119883

such that 1198911198991199110

rarr 119905 Since (1198911198991199110 119891119899+1

1199110) isin 119864(119866) for all

119899 isin N and 119866 is a (119862119891) graphs there exists a subsequence

1198911198991198961199110 of 119891

1198991199110 and 119901 isin N such that (119891

1198991198961199110 119905) isin 119864(119866) for

all 119896 ge 119901 Observe that (1199110 1198911199110 11989121199110 119891

11989911199110 119891

1198991199011199110 119905)

is a path in 119866 Therefore 119905 isin [1199110]119866 From (2) we get

119889 (119891119899119896+11199110 119891119905) le 120593 (119889 (119891

1198991198961199110 119905)) lt 119889 (119891

1198991198961199110 119905) forall119896 ge 119899

0

(14)

Letting 119896 rarr infin we obtain lim119896rarrinfin

119891119899119896+11199110

= 119891119905 as 119889

is continuous Since 1198911198991198961199110 is a subsequence of 119891

1198991199110 we

conclude that 119891119905 = 119905 Finally if 119910 isin [1199110]119866 it follows from

Lemma 12 that 119891119899119910 rarr 119905(2) Let 119866 be weakly connected (119867

119891)-graph From

Proposition 13 1198911198991199110

rarr 119905 isin 119883 and then 119903(1198911198991199110 119905) rarr 0 as

119899 rarr infin Now for each 119899 isin N let 119910119899

119894 119894 = 0 1 119872

119899be a

path from 1198911198991199110to 119905 with 119910

0= 119905 and 119910

119899

119872119899

= 1198911198991199110in 119866 then

119889 (119905 119891119905) le 119904 [119889 (119905 119891119899+1

1199110) + 119889 (119891

119899+1

1199110 119891119905)]

le 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

119889 (119891119910119899

119894minus1 119891119910119899

119894)]

le 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

120593 (119889 (119910119899

119894minus1 119910119899

119894))]

lt 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

119889 (119910119899

119894minus1 119910119899

119894)]

= 119904 [119889 (119905 119891119899+1

1199110) + 119903 (119891

119899

1199110 119905)]

(15)

Letting 119899 rarr infin the above inequality yields 119891119905 = 119905 Let 119910 isin

[1199110]119866

= 119883 be arbitrary then from Lemma 12 and Remark 10it is easily seen that 119891119899119910 rarr 119905

The following example shows that the condition of (119862119891)

or (119867119891)-graph in the hypothesis of Theorem 18 can not be

dropped

Example 19 Let119883 = [0 1] let 119889(119909 119910) = |119909minus119910|2 and let119891119909 =

1199092 for all 119909 isin (0 1] and 1198910 = 12 Then (119883 119889) is a complete119887-metric space with 119904 = 2 Further 119889 is continuous and 119891 isa 119887-(120593 119866

1) contraction (with 120593(119905) = 1199054) where 119881(119866

1) = 119883

and 119864(1198661) = (119909 119910) isin (0 1] times (0 1] 119909 ge 119910 cup (0 0) (0 1)

Note that 1198661is weakly connected but 119891 has no fixed point in

[1199110]1198661

= 119883 Observe that 1198661is not a (119862

119891)-graph because the

sequence 119891119899119909 = 1199092

119899rarr 0 for 119909 isin (0 1] and (119891

119899119909 119891119899+1

119909) isin

119864(1198661) 119899 isin N but it does not contain any subsequence such

that (119909119899119896

0) isin 119864(1198661) Also we note that for any fixed 119909 isin

(0 1] 119903(119891119899119909 0) = 2[|1199092119899minus 1|2

+ |1 minus 0|2

] 999424999426999456 0 as 119899 rarr infin

Definition 20 Let (119883 119889) be a 119887-metric space A mapping 119891

119883 rarr 119883 is called orbitally continuous if for all 119909 119910 isin 119883 andany sequence 119896

119899119899isinN of positive integers 119891119896119899119909 rarr 119910 implies

119891(119891119896119899119909) rarr 119891119910 as 119899 rarr infin A mapping 119891 119883 rarr 119883 is called

orbitally 119866-continuous if for all 119909 119910 isin 119883 and any sequence119896119899119899isinN of positive integers 119891119896119899119909 rarr 119910 and (119891

119896119899119909 119891119896119899+1119909) isin

119864(119866) for all 119899 isin N imply 119891(119891119896119899119909) rarr 119891119910

Theorem21 Let (119883 119889) be a complete 119887-metric space and let119891be a 119887-(120593 119866) contraction where 120593 is a 119887-comparison functionAssume that 119889 is continuous 119891 is orbitally 119866-continuous andthere is 119911

0in119883 for which (119911

0 1198911199110) is an edge in 119866 Then 119891 has

a fixed point 119905 isin 119883 Moreover for any 119910 isin [1199110]119866 119891119899119910 rarr 119905

Proof It follows from Proposition 13 that 1198911198991199110 is a Cauchy

sequence in (119883 119889) Since 119883 is complete there exists 119905 isin 119883

such that lim119899rarrinfin

1198911198991199110

= 119905 Since (1198911198991199110 119891119899+1

1199110) isin 119864(119866) for

all 119899 isin N 119891 is orbitally 119866-continuous Therefore continuityof 119889 implies that 119891119905 = 119905 Let 119910 isin [119911

0]119866be arbitrary then it

follows from Lemma 12 that lim119899rarrinfin

119891119899119910 = 119905

Abstract and Applied Analysis 5

Slightly strengthening the continuity condition on 119891 ournext theorem deals with the graph 119866 which may fail to havethe property that there is 119911

0in119883 for which (119911

0 1198911199110) is an edge

in 119866

Theorem22 Let (119883 119889) be a complete 119887-metric space and let119891be a 119887-(120593 119866) contraction where 120593 is a 119887-comparison functionAssume that 119889 is continuous 119891 is orbitally continuous andthere is 119911

0in119883 for which 119891119911

0isin [1199110]119866 Then for any 119910 isin [119911

0]119866

119891119899119910 rarr 119905 isin 119883 where 119905 is a fixed point of 119891

Proof It follows from Proposition 13 that 1198911198991199110 is a Cauchy

sequence in 119883 Since 119883 is complete there exists 119905 isin 119883

such that 1198911198991199110

rarr 119905 Since 119891 is orbitally continuous thenlim119899rarrinfin

1198911198911198991199110

= 119891119905 which yields 119891119905 = 119905 Let 119910 isin [1199110]119866be

arbitrary then from Lemma 12 lim119899rarrinfin

119891119899119910 = 119905

Remark 23 In addition to the hypothesis ofTheorems 21 and22 if we assume that 119866 is weakly connected then 119891 willbecome Picard operator [11] on 119883

Remark 24 Theorem 18 generalizesextends claims 40 and

50 of [10 Theorem 32] and [7 Theorem 4(1)] Theorem 21generalizes claims 20 and 3

0 of [10Theorem 33]Theorem 22generalizes claims 2

0 and 30 of [10 Theorem 34] and thus

generalizes extends results ofNieto andRodrıguez-Lopez [24Theorems 21 and 23] Petrusel andRus [11Theorem43] andRan and Reurings [25 Theorem 21] We mention here thatTheorem 18 can not be improved using comparison functioninstead of 119887-comparison function (see Gwozdz-Łukawskaand Jachymski [26 Example 2])

We observe that Theorem 22 can be used to extendfamous fixed point theorem of Edelstein to the case of119887-metric space We need to define notion of 120598-chainableproperty for 119887-metric space

Definition 25 A 119887-metric space (119883 119889) is said to be 120598-chainable for some 120598 gt 0 if for119909 119910 isin 119883 there exist119909

119894isin 119883 119894 =

0 1 2 119897 with 1199090

= 119909 119909119897= 119910 such that 119889(119909

119894minus1 119909119894) lt 120598 for

119894 = 1 2 119897

Corollary 26 Let (119883 119889) be a complete 120598-chainable 119887-metricspace Assume that 119889 is continuous and there exists a 119887-comparison function 120593 R+ rarr R+ such that 119891 119883 rarr 119883

satisfying

119889 (119909 119910) lt 120598 119894119898119901119897119894119890119904 119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) (16)

for all 119909 119910 isin 119883 Then 119891 is a Picard operator

Proof Consider a graph 119866 consisting of 119881(119866) = 119883 and(119909 119910) isin 119864(119866) if and only if 119889(119909 119910) lt 120598 Since119883 is 120598-chainable119866 is weakly connected Let (119909 119910) isin 119864(119866) and from (16) wehave

119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) lt 119889 (119909 119910) lt 120598 (17)

Then 119889(119891119909 119891119910) isin 119864(119866) Therefore in view of (16) 119891 is 119887-(120593 119866) contraction Further (16) implies that 119891 is continuousNow the conclusion follows by usingTheorem 22

Now we establish a fixed point theorem using a generalcontractive condition

Theorem 27 Let (119883 119889) be a complete 119887-metric space and let119866 be a (119862

119891)-graph in119883times119883 such that119881(119866) = 119883 and 119891 119883 rarr

119883 is an edge-preserving mapping Assume that 119889 is continuousand there exist 120572 120573 120574 ge 0 with 119904120572 + (119904 + 1)120573 + 119904(119904 + 1)120574 lt 1

and for all (119909 119910) isin 119864(119866)

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(18)

If there is 1199110in119883 for which (119911

0 1198911199110) is an edge in119866 then119891 has

a fixed point in [1199110]119866

Proof Since 119891 is edge-preserving then (1198911198991199110 119891119899+1

1199110) isin

119864(119866) for all 119899 isin N From (18) and using (d3) it follows that

119889 (119891119899

1199110 119891119899+1

1199110)

le 120572119889 (119891119899minus1

1199110 119891119899

1199110)

+ 120573 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1z0)]

+ 120574119904 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1

1199110)]

(19)

On rearranging

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904] 119889 (119891

119899minus1

1199110 119891119899

1199110) (20)

Repeating iteratively we have

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119899

119889 (1199110 1198911199110) (21)

For 119898 gt 119899 ge 1 and using (d3) Definition 1 we have

119889 (119891119899

1199110 119891119898

1199110)

le 119904119889 (119891119899

1199110 119891119899+1

1199110) + 1199042

119889 (119891119899+1

1199110 119891119899+2

1199110)

+ sdot sdot sdot + 119904119898minus119899

119889 (119891119898minus1

1199110 119891119898

1199110)

le1

119904119899minus1119889 (1199110 1198911199110)

119898minus1

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(using (21))

lt1

119904119899minus1119889 (1199110 1198911199110)

infin

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(22)

Since 119904[(120572 + 120573 + 120574119904)(1 minus 120573 minus 120574119904)] lt 1 then 1198911198991199110 is a

Cauchy sequence in 119883 By completeness of 119883 the sequence1198911198991199110 converges to 119905 isin 119883 Since 119866 is a (119862

119891)-graph there

exists a subsequence 1198911198991198961199110 and anatural number119901 such that

6 Abstract and Applied Analysis

(1198911198991198961199110 119905) isin 119864(119866) for all 119896 ge 119901 Using (18) for all 119896 ge 119901 we

have

119889 (119891119899119896+11199110 119891119905)

le 120572119889 (1198911198991198961199110 119905) + 120573 [119889 (119891

1198991198961199110 119891119899119896+11199110) + 119889 (119905 119891119905)]

+ 120574 [119889 (1198911198991198961199110 119891119905) + 119889 (119905 119891

119899119896+11199110)]

(23)

Since the 119887-metric 119889 is continuous and 120573 + 120574 lt 1 letting119896 rarr infin inequality (23) yields 119891119905 = 119905 Also note that(1199110 1198911199110 11989121199110 119891

11989911199110 119891

119899119901 119905) is a path in 119866 and hencein 119866 therefore 119905 isin [119911

0]119866

We note thatTheorem 27 does not guarantee the unique-ness of fixed point but this can be accomplished under someassumptions as in the following theorem

Theorem 28 In addition to the hypothesis ofTheorem 27 onefurther assumes that if 120572 + 2119904120573 + 2120574 lt 1 for the same set of120572 120573 120574 ge 0 and for any two fixed points 119905

1 1199052exist119911 isin 119883 such that

(1199051 119911) and (119905

2 119911) isin 119864(119866) Then 119891 has a unique fixed point

Proof Let 1199051and 1199052be two fixed points of 119891 and then there

exists 119911 isin 119883 such that (1199051 119911) (119905

2 119911) isin 119864(119866) By induction we

have (1199051 119891119899119911) (1199052 119891119899119911) isin 119864(119866) for all 119899 = 0 1 From (18)

we have

119889 (1199051 119891119899

119911) le 120572119889 (1199051 119891119899minus1

119911) + 120573119889 (119891119899minus1

119911 119891119899

119911)

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

le 120572119889 (1199051 119891119899minus1

119911) + 120573119904 [119889 (119891119899minus1

119911 1199051) + 119889 (119905

1 119891119899

119911)]

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

(24)

On rearranging

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574] 119889 (1199051 119891119899minus1

119911) forall119899 = 1 2

(25)

Continuing recursively (25) gives

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574]

119899

119889 (1199051 119911) (26)

Since [(120572+119904120573+120574)(1minus119904120573minus120574)] lt 1 then lim119899rarrinfin

119889(1199051 119891119899119911) =

0 Similarly one can show that lim119899rarrinfin

119889(1199052 119891119899119911) = 0 which

by using (d3) Definition 1 infers that 119889(1199051 1199052) = 0

Suppose that (119883 ⪯) is a partially ordered set Considergraph 119866

2consisting of 119864(119866

2) = (119909 119910) isin 119883 times 119883 119909 ⪯

119910 or119910 ⪯ 119909 and 119881(1198662) coincides with 119883 We note that

if a self-mapping 119891 is monotone with respect to the orderthen for graph 119866

2 it is obvious that 119891 is edge-preserving

or equivalently we can say that 119891 maps comparable elementsonto comparable elements

Following corollaries are the direct consequences ofTheorem 28

Corollary 29 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists 120572 120573 120574 ge 0 with 120572 + 2120573 + 2120574 lt 1 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(27)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 30 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (28)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 31 Let (119883 119889) be a complete metric space where 119883

is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (29)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Remark 32 Wenote that inTheorem 27 the condition ldquothereis 1199110in 119883 for which (119911

0 1198911199110) is an edge in 119866rdquo yields 119891

1198991199110

rarr

119905 where 119905 isin 119883 is a fixed point of 119891 Consider a graph119866 = (119883119883 times 119883) For such graph under the assumptions ofTheorems 27 and 28 it infers that119891 is a Picard operatorThusmany standard fixed point theorems can be easily deducedfromTheorem 28 as follows

Abstract and Applied Analysis 7

Corollary 33 (Hardy and Rogers [27]) Let (119883 119889) be acomplete metric space and let 119891 119883 rarr 119883 Suppose that thereexists constants 120572 120573 120574 ge 0 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(30)

for all 119909 119910 isin 119883 where 120572 + 2120573 + 2120574 lt 1 then 119891 has a uniquefixed point in 119883

Corollary 34 (Kannan [28]) Let (119883 119889) be a complete metricspace and let119891 119883 rarr 119883 Suppose that there exists a constants119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (31)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

Corollary 35 (Chatterjea [29]) Let (119883 119889) be a completemetric space and let 119891 119883 rarr 119883 Suppose that there existsa constants 119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (32)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

3 Applications

Let 119883 be a nonempty set let 119898 be a positive integer 119883119894119898

119894=1

be nonempty closed subsets of 119883 and let 119891 ⋃119898

119894=1119883119894

rarr

⋃119898

119894=1119883119894be an operatorThen119883 = ⋃

119898

119894=1119883119894is known as cyclic

representation of 119883 wrt 119891 if

119891 (1198831) sub 119883

2 119891 (119883

119898minus1) sub 119883

119898 119891 (119883

119898) sub 119883

1

(33)

and operator 119891 is known as cyclic operator [30]

Theorem 36 Let (119883 119889) be complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

120593 R+ rarr R+ be a 119887-comparison function and let 119891 119884 rarr

119884 Further suppose that(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) 119889(119891119909 119891119910) le 120593(119889(119909 119910)) whenever 119909 isin 119883119894 119910 isin 119883

119894+1

where 119883119898+1

= 1198831

Then 119891 has a unique fixed point 119905 isin ⋂119898

119894=1119883119894and 119891

119899119910 rarr 119905 for

any 119910 isin ⋃119898

119894=1119883119894

Proof We note that (119884 119889) is complete 119887-metric space Let usconsider a graph 119866 consisting of 119881(119866) = 119884 and 119864(119866) =

Δ cup (119909 119910) isin 119884 times 119884 119909 isin 119883119894 119910 isin 119883

119894+1 119894 = 1 119898 By

(i) it follows that 119891 preserves edges Now let 119891119899119909 rarr 119909

lowast in119884 such that (119891119899119909 119891

119899+1119909) isin 119864(119866) for all 119899 ge 1 then in view of

(33) sequence 119891119899119909 has infinitely many terms in each 119883

119894so

that one can easily extract a subsequence of 119891119899119909 convergingto 119909lowast in each 119883

119894 Since 119883

119894rsquos are closed then 119909

lowastisin ⋂119898

119894=1119883119894

Now it is easy to form a subsequence 119891119899119896119909 in some 119883

119895

119895 isin 1 119898 such that (119891119899119896119909 119909lowast) isin 119864(119866) for 119896 ge 1 and

it indicates that 119866 is weakly connected (119862119891)-graph and thus

conclusion follows fromTheorem 18

Remark 37 (see [31Theorem 21(1)]) It can be deduced fromTheorem 36 if (119883 119889) is a metric space

On the same lines as in proof of Theorem 36 we obtainthe following consequence of Theorem 28

Theorem38 Let (119883 119889) be a complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

and let 119891 119884 rarr 119884 Further suppose that

(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) there exist 120572 120573 120574 ge 0with 119904120572+119904(119904+1)120573+119904(119904+1)120574 lt 1

such that

119889 (119891119909 119891119910)

le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(34)

whenever 119909 isin 119883119894 119910 isin 119883

119894+1 where 119883

119898+1= 1198831

Then 119891 has a fixed point 119905 isin ⋂119898

119894=1119883119894

Remark 39 [32 Theorem 7] and [33 Theorem 31] can bededuced from Theorem 38 but it does not ensure 119891 to be aPicard operator

Remark 40 We note that in proof [32 Theorem 7] theauthorrsquos argument to prove that is 119866 (as assumed in proofof Theorem 36) a (119862)-graph is valid only if the terms ofsequence 119909

119899 are Picard iterations otherwise it is void For

example let 119884 = ⋃3

119894=1119883119894where 119883

1= 1119899 119899 is odd cup 0

1198832= 1119899 119899 is even cup 0 119883

3= 0 and define 119891 119884 rarr 119884

as 119891119909 = 1199092 119909 isin 119884 1198832and 119891119909 = 0 119909 isin 119883

2 We see that (33)

is satisfied and 1119899 rarr 0 but 1198833does not contain infinitely

many terms of 119909119899 In the following we give the corrected

argument to prove that 119866 is a (119862)-graph

Let 119909119899

rarr 119909 in 119883 such that (119909119899 119909119899+1

) isin 119864(119866) for all119899 ge 1 Keeping in mind construction of119866 there exists at leastone pair of closed sets 119883

119895 119883119895+1

for some 119895 isin 1 2 119898

such that both sets contain infinitely many terms of sequence119909119899 since 119883

119894rsquos are closed so that 119909 isin 119883

119895cap 119883119895+1

for some119895 isin 1 119898 and thus one can easily extract a subsequencesuch that (119909

119899119896 119909) isin 119864(119866) holds for 119896 ge 1

Now we establish an existence theorem for the solutionof an integral equation as a consequence of our Theorem 18

Theorem 41 Consider the integral equation

119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (35)

where 119896 [119886 119887] times [119886 119887] times R119899 rarr R119899 and 119892 [119886 119887] rarr R119899 iscontinuous Assume that

(i) 119896(119905 119904 sdot) R119899 rarr R119899 is nondecreasing for each 119905 119904 isin

[119886 119887](ii) there exists a (119888)-comparison function 120593 R+ rarr R+

and a continuous function 119901 [119886 119887] times [119886 119887] rarr R+

8 Abstract and Applied Analysis

such that |119896(119905 119904 119909(119904)) minus 119896(119905 119904 119910(119904))| le 119901(119905 119904)120593(|119909(119904) minus

119910(119904)|) for each 119905 119904 isin [119886 119887] and 119909 le 119910 (ie 119909(119905) le 119910(119905)

for all 119905 isin [119886 119887])

(iii) sup119905isin[119886119887]

int119887

119886119901(119905 119904)119889119904 le 1

(iv) there exists 1199090

isin 119862([119886 119887]R119899) such that 1199090(119905) le

int119887

119886119896(119905 119904 119909

0(119904))119889119904 + 119892(119905) for all 119905 isin [119886 119887]

Then the integral equation (35) has a unique solution in theset 119909 isin 119862([119886 119887]R119899) 119909(119905) le 119909

0(119905) or 119909(119905) ge 119909

0(119905) for all

119905 isin [119886 119887]

Proof Let (119883 sdot infin

) where 119883 = 119862([119886 119887]R119899) and define amapping 119879 119862([119886 119887]R119899) rarr 119862([119886 119887]R119899) by

119879119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (36)

Consider a graph 119866 consisting of 119881(119866) = 119883 and 119864(119866) =

(119909 119910) isin 119883 times 119883 119909(119905) le 119910(119905) for all 119905 isin [119886 119887] From property(i) we observe that the mapping 119879 is nondecreasing thus 119879

preserves edges Furthermore 119866 is a (119862)-graph that is forevery nondecreasing sequence 119909

119899 sub 119883 which converges to

some 119911 isin 119883 then 119909119899(119905) le 119911(119905) for all 119905 isin [119886 119887] Now for every

119909 119910 isin 119883 with (119909 119910) isin 119864(119866) we have

1003816100381610038161003816119879119909 (119905) minus 119879119910 (119905)1003816100381610038161003816

le int

119887

119886

1003816100381610038161003816119896 (119905 119904 119909 (119904)) minus 119896 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904

le int

119887

119886

119901 (119905 119904) 120593 (1003816100381610038161003816119909 (119904) minus 119910 (119904)

1003816100381610038161003816) 119889119904

le 120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817infin) int

119887

119886

119901 (119905 119904) 119889119904

(37)

Hence 119889(119879119909 119879119910) le 120593(119889(119909 119910)) From (iv) we have(1199090 1198791199090) isin 119864(119866) so that [119909

0]119866

= 119909 isin 119862([119886 119887]R119899) 119909(119905) le

1199090(119905) or 119909(119905) ge 119909

0(119905) for all 119905 isin [119886 119887] The conclusion follows

fromTheorem 18

Note that Theorem 41 specifies region of solution whichinvokes the novelty of our result

References

[1] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacesrdquo Journal of Functional Analysis vol 30 pp 26ndash37 1989

[2] J Heinonen Lectures on Analysis on Metric Spaces SpringerNew York NY USA 2001

[3] N Bourbaki Topologie Generale Herman Paris France 1974[4] S Czerwik ldquoNonlinear set-valued contraction mappings in

119887-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[5] M Bota A Molnar and C Varga ldquoOn Ekelandrsquos variationalprinciple in 119887-metric spacesrdquo Fixed Point Theory vol 12 no 1pp 21ndash28 2011

[6] M F Bota-Boriceanu andA Petrusel ldquoUlam-Hyers stability foroperatorial equationsrdquo Analele Stiintifice ale Universitatii ldquoAl ICuzardquo din Iasi vol 57 supplement 1 pp 65ndash74 2011

[7] M Pacurar ldquoA fixed point result for 120593-contractions on 119887-metricspaces without the boundedness assumptionrdquo PolytechnicaPosnaniensis InstitutumMathematicum Fasciculi Mathematicino 43 pp 127ndash137 2010

[8] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis Babes-Bolyai vol 41 no 4pp 23ndash27 1996

[9] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[10] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[11] A Petrusel and I A Rus ldquoFixed point theorems in ordered l-spacesrdquo Proceedings of the American Mathematical Society vol134 no 2 pp 411ndash418 2006

[12] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 p 68 1975

[13] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[14] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008

[15] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[16] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997

[17] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquoSeminar on Fixed Point Theory vol 3 pp 3ndash9 1993

[18] T P Petru and M Boriceanu ldquoFixed point results for gener-alized 120593-contraction on a set with two metricsrdquo TopologicalMethods in Nonlinear Analysis vol 33 no 2 pp 315ndash326 2009

[19] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 11 pages 2012

[20] A Amini-Harandi ldquoCoupled and tripled fixed point theory inpartially ordered metric spaces with application to initial valueproblemrdquoMathematical and Computer Modelling vol 57 no 9-10 pp 2343ndash2348 2012

[21] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[22] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012

[23] M Samreen and T Kamran ldquoFixed point theorems for integralG-contractionsrdquo Fixed Point Theory and Applications vol 2013article 149 2013

[24] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[25] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrix

Abstract and Applied Analysis 9

equationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[26] G Gwozdz-Łukawska and J Jachymski ldquoIFS on a metric spacewith a graph structure and extensions of the Kelisky-Rivlintheoremrdquo Journal of Mathematical Analysis and Applicationsvol 356 no 2 pp 453ndash463 2009

[27] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973

[28] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968

[29] S K Chatterjea ldquoFixed-point theoremsrdquo Doklady BolgarskoıAkademii Nauk vol 25 pp 727ndash730 1972

[30] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003

[31] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010

[32] F Bojor ldquoFixed point theorems for Reich type contractionson metric spaces with a graphrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 9 pp 3895ndash3901 2012

[33] M A Petric ldquoSome remarks concerning Ciric-Reich-Rus oper-atorsrdquo Creative Mathematics and Informatics vol 18 no 2 pp188ndash193 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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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 Abstract and Applied Analysis

Example 16 Let 119883 = 1119899 119899 isin N cup 0 cup N with respect to119887-metric 119889(119909 119910) = |119909 minus 119910|

2 and let 119868 be identity map on 119883Consider a graph 119866

2consisting of 119881(119866

2) = 119883 and

119864 (1198662) = (

1

119899

1

119899 + 1) (

1

119899 + 1 119899) (119899 0)

(1

5119899 0) 119899 isin N

(12)

since 119909119899

= 1119899 rarr 0 as 119899 rarr infin We note that 1198662is a (119862

119868)-

graph but 119903(119909119899 0) = 2|(1119899) minus (1(119899 + 1))|

2+ 22|(1(119899 + 1)) minus

119899|2+ 221198992999424999426999456 0 as 119899 rarr infin Thus 119866

2is not an (119867

119868)-graph

Example 17 Let119883 = 1119899 119899 isin Ncupradic5119899 119899 isin Ncup0withrespect to 119887-metric119889(119909 119910) = |119909minus119910|

2 and let 119868 be identitymapon 119883 Consider a graph 119866

3consisting of 119881(119866

3) = 119883 and

119864 (1198663) = (

1

119899

1

119899 + 1) (

1

119899 + 1radic5

119899) (

radic5

119899 0) 119899 isin N

(13)

since 119909119899

= 1119899 rarr 0 as 119899 rarr infin Clearly 1198663is not a (119862

119868)-

graph but it is easy to verify that 1198663is an (119867

119868)-graph

Above examples show that for a given 119891 notions (119862119891)-

graph and (119867119891)-graph are independent even if 119891 is identity

map

Theorem 18 Let (119883 119889) be a complete 119887-metric space and let119891 be a 119887-(120593 119866) contraction where 120593 is 119887-comparison functionAssume that 119889 is continuous and there is 119911

0in 119883 for which

(1199110 1198911199110) is an edge in 119866 Then the following assertions hold

(1) If 119866 is a (119862119891)-graph then 119891 has a unique fixed point

119905 isin [1199110]119866 and for any 119910 isin [119911

0]119866 119891119899119910 rarr 119905 Further if

119866 is weakly connected then 119891 is Picard operator

(2) If 119866 is weakly connected (119867119891)-graph then 119891 has a

unique fixed point 119905 isin 119883 and for any 119910 isin 119883 119891119899119910 rarr 119905

Proof (1) It follows from Proposition 13 that 1198911198991199110 is a

Cauchy sequence in119883 Since119883 is complete there exists 119905 isin 119883

such that 1198911198991199110

rarr 119905 Since (1198911198991199110 119891119899+1

1199110) isin 119864(119866) for all

119899 isin N and 119866 is a (119862119891) graphs there exists a subsequence

1198911198991198961199110 of 119891

1198991199110 and 119901 isin N such that (119891

1198991198961199110 119905) isin 119864(119866) for

all 119896 ge 119901 Observe that (1199110 1198911199110 11989121199110 119891

11989911199110 119891

1198991199011199110 119905)

is a path in 119866 Therefore 119905 isin [1199110]119866 From (2) we get

119889 (119891119899119896+11199110 119891119905) le 120593 (119889 (119891

1198991198961199110 119905)) lt 119889 (119891

1198991198961199110 119905) forall119896 ge 119899

0

(14)

Letting 119896 rarr infin we obtain lim119896rarrinfin

119891119899119896+11199110

= 119891119905 as 119889

is continuous Since 1198911198991198961199110 is a subsequence of 119891

1198991199110 we

conclude that 119891119905 = 119905 Finally if 119910 isin [1199110]119866 it follows from

Lemma 12 that 119891119899119910 rarr 119905(2) Let 119866 be weakly connected (119867

119891)-graph From

Proposition 13 1198911198991199110

rarr 119905 isin 119883 and then 119903(1198911198991199110 119905) rarr 0 as

119899 rarr infin Now for each 119899 isin N let 119910119899

119894 119894 = 0 1 119872

119899be a

path from 1198911198991199110to 119905 with 119910

0= 119905 and 119910

119899

119872119899

= 1198911198991199110in 119866 then

119889 (119905 119891119905) le 119904 [119889 (119905 119891119899+1

1199110) + 119889 (119891

119899+1

1199110 119891119905)]

le 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

119889 (119891119910119899

119894minus1 119891119910119899

119894)]

le 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

120593 (119889 (119910119899

119894minus1 119910119899

119894))]

lt 119904 [119889 (119905 119891119899+1

1199110) +

119872119899

sum

119894=1

119904119894

119889 (119910119899

119894minus1 119910119899

119894)]

= 119904 [119889 (119905 119891119899+1

1199110) + 119903 (119891

119899

1199110 119905)]

(15)

Letting 119899 rarr infin the above inequality yields 119891119905 = 119905 Let 119910 isin

[1199110]119866

= 119883 be arbitrary then from Lemma 12 and Remark 10it is easily seen that 119891119899119910 rarr 119905

The following example shows that the condition of (119862119891)

or (119867119891)-graph in the hypothesis of Theorem 18 can not be

dropped

Example 19 Let119883 = [0 1] let 119889(119909 119910) = |119909minus119910|2 and let119891119909 =

1199092 for all 119909 isin (0 1] and 1198910 = 12 Then (119883 119889) is a complete119887-metric space with 119904 = 2 Further 119889 is continuous and 119891 isa 119887-(120593 119866

1) contraction (with 120593(119905) = 1199054) where 119881(119866

1) = 119883

and 119864(1198661) = (119909 119910) isin (0 1] times (0 1] 119909 ge 119910 cup (0 0) (0 1)

Note that 1198661is weakly connected but 119891 has no fixed point in

[1199110]1198661

= 119883 Observe that 1198661is not a (119862

119891)-graph because the

sequence 119891119899119909 = 1199092

119899rarr 0 for 119909 isin (0 1] and (119891

119899119909 119891119899+1

119909) isin

119864(1198661) 119899 isin N but it does not contain any subsequence such

that (119909119899119896

0) isin 119864(1198661) Also we note that for any fixed 119909 isin

(0 1] 119903(119891119899119909 0) = 2[|1199092119899minus 1|2

+ |1 minus 0|2

] 999424999426999456 0 as 119899 rarr infin

Definition 20 Let (119883 119889) be a 119887-metric space A mapping 119891

119883 rarr 119883 is called orbitally continuous if for all 119909 119910 isin 119883 andany sequence 119896

119899119899isinN of positive integers 119891119896119899119909 rarr 119910 implies

119891(119891119896119899119909) rarr 119891119910 as 119899 rarr infin A mapping 119891 119883 rarr 119883 is called

orbitally 119866-continuous if for all 119909 119910 isin 119883 and any sequence119896119899119899isinN of positive integers 119891119896119899119909 rarr 119910 and (119891

119896119899119909 119891119896119899+1119909) isin

119864(119866) for all 119899 isin N imply 119891(119891119896119899119909) rarr 119891119910

Theorem21 Let (119883 119889) be a complete 119887-metric space and let119891be a 119887-(120593 119866) contraction where 120593 is a 119887-comparison functionAssume that 119889 is continuous 119891 is orbitally 119866-continuous andthere is 119911

0in119883 for which (119911

0 1198911199110) is an edge in 119866 Then 119891 has

a fixed point 119905 isin 119883 Moreover for any 119910 isin [1199110]119866 119891119899119910 rarr 119905

Proof It follows from Proposition 13 that 1198911198991199110 is a Cauchy

sequence in (119883 119889) Since 119883 is complete there exists 119905 isin 119883

such that lim119899rarrinfin

1198911198991199110

= 119905 Since (1198911198991199110 119891119899+1

1199110) isin 119864(119866) for

all 119899 isin N 119891 is orbitally 119866-continuous Therefore continuityof 119889 implies that 119891119905 = 119905 Let 119910 isin [119911

0]119866be arbitrary then it

follows from Lemma 12 that lim119899rarrinfin

119891119899119910 = 119905

Abstract and Applied Analysis 5

Slightly strengthening the continuity condition on 119891 ournext theorem deals with the graph 119866 which may fail to havethe property that there is 119911

0in119883 for which (119911

0 1198911199110) is an edge

in 119866

Theorem22 Let (119883 119889) be a complete 119887-metric space and let119891be a 119887-(120593 119866) contraction where 120593 is a 119887-comparison functionAssume that 119889 is continuous 119891 is orbitally continuous andthere is 119911

0in119883 for which 119891119911

0isin [1199110]119866 Then for any 119910 isin [119911

0]119866

119891119899119910 rarr 119905 isin 119883 where 119905 is a fixed point of 119891

Proof It follows from Proposition 13 that 1198911198991199110 is a Cauchy

sequence in 119883 Since 119883 is complete there exists 119905 isin 119883

such that 1198911198991199110

rarr 119905 Since 119891 is orbitally continuous thenlim119899rarrinfin

1198911198911198991199110

= 119891119905 which yields 119891119905 = 119905 Let 119910 isin [1199110]119866be

arbitrary then from Lemma 12 lim119899rarrinfin

119891119899119910 = 119905

Remark 23 In addition to the hypothesis ofTheorems 21 and22 if we assume that 119866 is weakly connected then 119891 willbecome Picard operator [11] on 119883

Remark 24 Theorem 18 generalizesextends claims 40 and

50 of [10 Theorem 32] and [7 Theorem 4(1)] Theorem 21generalizes claims 20 and 3

0 of [10Theorem 33]Theorem 22generalizes claims 2

0 and 30 of [10 Theorem 34] and thus

generalizes extends results ofNieto andRodrıguez-Lopez [24Theorems 21 and 23] Petrusel andRus [11Theorem43] andRan and Reurings [25 Theorem 21] We mention here thatTheorem 18 can not be improved using comparison functioninstead of 119887-comparison function (see Gwozdz-Łukawskaand Jachymski [26 Example 2])

We observe that Theorem 22 can be used to extendfamous fixed point theorem of Edelstein to the case of119887-metric space We need to define notion of 120598-chainableproperty for 119887-metric space

Definition 25 A 119887-metric space (119883 119889) is said to be 120598-chainable for some 120598 gt 0 if for119909 119910 isin 119883 there exist119909

119894isin 119883 119894 =

0 1 2 119897 with 1199090

= 119909 119909119897= 119910 such that 119889(119909

119894minus1 119909119894) lt 120598 for

119894 = 1 2 119897

Corollary 26 Let (119883 119889) be a complete 120598-chainable 119887-metricspace Assume that 119889 is continuous and there exists a 119887-comparison function 120593 R+ rarr R+ such that 119891 119883 rarr 119883

satisfying

119889 (119909 119910) lt 120598 119894119898119901119897119894119890119904 119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) (16)

for all 119909 119910 isin 119883 Then 119891 is a Picard operator

Proof Consider a graph 119866 consisting of 119881(119866) = 119883 and(119909 119910) isin 119864(119866) if and only if 119889(119909 119910) lt 120598 Since119883 is 120598-chainable119866 is weakly connected Let (119909 119910) isin 119864(119866) and from (16) wehave

119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) lt 119889 (119909 119910) lt 120598 (17)

Then 119889(119891119909 119891119910) isin 119864(119866) Therefore in view of (16) 119891 is 119887-(120593 119866) contraction Further (16) implies that 119891 is continuousNow the conclusion follows by usingTheorem 22

Now we establish a fixed point theorem using a generalcontractive condition

Theorem 27 Let (119883 119889) be a complete 119887-metric space and let119866 be a (119862

119891)-graph in119883times119883 such that119881(119866) = 119883 and 119891 119883 rarr

119883 is an edge-preserving mapping Assume that 119889 is continuousand there exist 120572 120573 120574 ge 0 with 119904120572 + (119904 + 1)120573 + 119904(119904 + 1)120574 lt 1

and for all (119909 119910) isin 119864(119866)

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(18)

If there is 1199110in119883 for which (119911

0 1198911199110) is an edge in119866 then119891 has

a fixed point in [1199110]119866

Proof Since 119891 is edge-preserving then (1198911198991199110 119891119899+1

1199110) isin

119864(119866) for all 119899 isin N From (18) and using (d3) it follows that

119889 (119891119899

1199110 119891119899+1

1199110)

le 120572119889 (119891119899minus1

1199110 119891119899

1199110)

+ 120573 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1z0)]

+ 120574119904 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1

1199110)]

(19)

On rearranging

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904] 119889 (119891

119899minus1

1199110 119891119899

1199110) (20)

Repeating iteratively we have

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119899

119889 (1199110 1198911199110) (21)

For 119898 gt 119899 ge 1 and using (d3) Definition 1 we have

119889 (119891119899

1199110 119891119898

1199110)

le 119904119889 (119891119899

1199110 119891119899+1

1199110) + 1199042

119889 (119891119899+1

1199110 119891119899+2

1199110)

+ sdot sdot sdot + 119904119898minus119899

119889 (119891119898minus1

1199110 119891119898

1199110)

le1

119904119899minus1119889 (1199110 1198911199110)

119898minus1

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(using (21))

lt1

119904119899minus1119889 (1199110 1198911199110)

infin

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(22)

Since 119904[(120572 + 120573 + 120574119904)(1 minus 120573 minus 120574119904)] lt 1 then 1198911198991199110 is a

Cauchy sequence in 119883 By completeness of 119883 the sequence1198911198991199110 converges to 119905 isin 119883 Since 119866 is a (119862

119891)-graph there

exists a subsequence 1198911198991198961199110 and anatural number119901 such that

6 Abstract and Applied Analysis

(1198911198991198961199110 119905) isin 119864(119866) for all 119896 ge 119901 Using (18) for all 119896 ge 119901 we

have

119889 (119891119899119896+11199110 119891119905)

le 120572119889 (1198911198991198961199110 119905) + 120573 [119889 (119891

1198991198961199110 119891119899119896+11199110) + 119889 (119905 119891119905)]

+ 120574 [119889 (1198911198991198961199110 119891119905) + 119889 (119905 119891

119899119896+11199110)]

(23)

Since the 119887-metric 119889 is continuous and 120573 + 120574 lt 1 letting119896 rarr infin inequality (23) yields 119891119905 = 119905 Also note that(1199110 1198911199110 11989121199110 119891

11989911199110 119891

119899119901 119905) is a path in 119866 and hencein 119866 therefore 119905 isin [119911

0]119866

We note thatTheorem 27 does not guarantee the unique-ness of fixed point but this can be accomplished under someassumptions as in the following theorem

Theorem 28 In addition to the hypothesis ofTheorem 27 onefurther assumes that if 120572 + 2119904120573 + 2120574 lt 1 for the same set of120572 120573 120574 ge 0 and for any two fixed points 119905

1 1199052exist119911 isin 119883 such that

(1199051 119911) and (119905

2 119911) isin 119864(119866) Then 119891 has a unique fixed point

Proof Let 1199051and 1199052be two fixed points of 119891 and then there

exists 119911 isin 119883 such that (1199051 119911) (119905

2 119911) isin 119864(119866) By induction we

have (1199051 119891119899119911) (1199052 119891119899119911) isin 119864(119866) for all 119899 = 0 1 From (18)

we have

119889 (1199051 119891119899

119911) le 120572119889 (1199051 119891119899minus1

119911) + 120573119889 (119891119899minus1

119911 119891119899

119911)

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

le 120572119889 (1199051 119891119899minus1

119911) + 120573119904 [119889 (119891119899minus1

119911 1199051) + 119889 (119905

1 119891119899

119911)]

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

(24)

On rearranging

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574] 119889 (1199051 119891119899minus1

119911) forall119899 = 1 2

(25)

Continuing recursively (25) gives

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574]

119899

119889 (1199051 119911) (26)

Since [(120572+119904120573+120574)(1minus119904120573minus120574)] lt 1 then lim119899rarrinfin

119889(1199051 119891119899119911) =

0 Similarly one can show that lim119899rarrinfin

119889(1199052 119891119899119911) = 0 which

by using (d3) Definition 1 infers that 119889(1199051 1199052) = 0

Suppose that (119883 ⪯) is a partially ordered set Considergraph 119866

2consisting of 119864(119866

2) = (119909 119910) isin 119883 times 119883 119909 ⪯

119910 or119910 ⪯ 119909 and 119881(1198662) coincides with 119883 We note that

if a self-mapping 119891 is monotone with respect to the orderthen for graph 119866

2 it is obvious that 119891 is edge-preserving

or equivalently we can say that 119891 maps comparable elementsonto comparable elements

Following corollaries are the direct consequences ofTheorem 28

Corollary 29 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists 120572 120573 120574 ge 0 with 120572 + 2120573 + 2120574 lt 1 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(27)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 30 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (28)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 31 Let (119883 119889) be a complete metric space where 119883

is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (29)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Remark 32 Wenote that inTheorem 27 the condition ldquothereis 1199110in 119883 for which (119911

0 1198911199110) is an edge in 119866rdquo yields 119891

1198991199110

rarr

119905 where 119905 isin 119883 is a fixed point of 119891 Consider a graph119866 = (119883119883 times 119883) For such graph under the assumptions ofTheorems 27 and 28 it infers that119891 is a Picard operatorThusmany standard fixed point theorems can be easily deducedfromTheorem 28 as follows

Abstract and Applied Analysis 7

Corollary 33 (Hardy and Rogers [27]) Let (119883 119889) be acomplete metric space and let 119891 119883 rarr 119883 Suppose that thereexists constants 120572 120573 120574 ge 0 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(30)

for all 119909 119910 isin 119883 where 120572 + 2120573 + 2120574 lt 1 then 119891 has a uniquefixed point in 119883

Corollary 34 (Kannan [28]) Let (119883 119889) be a complete metricspace and let119891 119883 rarr 119883 Suppose that there exists a constants119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (31)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

Corollary 35 (Chatterjea [29]) Let (119883 119889) be a completemetric space and let 119891 119883 rarr 119883 Suppose that there existsa constants 119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (32)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

3 Applications

Let 119883 be a nonempty set let 119898 be a positive integer 119883119894119898

119894=1

be nonempty closed subsets of 119883 and let 119891 ⋃119898

119894=1119883119894

rarr

⋃119898

119894=1119883119894be an operatorThen119883 = ⋃

119898

119894=1119883119894is known as cyclic

representation of 119883 wrt 119891 if

119891 (1198831) sub 119883

2 119891 (119883

119898minus1) sub 119883

119898 119891 (119883

119898) sub 119883

1

(33)

and operator 119891 is known as cyclic operator [30]

Theorem 36 Let (119883 119889) be complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

120593 R+ rarr R+ be a 119887-comparison function and let 119891 119884 rarr

119884 Further suppose that(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) 119889(119891119909 119891119910) le 120593(119889(119909 119910)) whenever 119909 isin 119883119894 119910 isin 119883

119894+1

where 119883119898+1

= 1198831

Then 119891 has a unique fixed point 119905 isin ⋂119898

119894=1119883119894and 119891

119899119910 rarr 119905 for

any 119910 isin ⋃119898

119894=1119883119894

Proof We note that (119884 119889) is complete 119887-metric space Let usconsider a graph 119866 consisting of 119881(119866) = 119884 and 119864(119866) =

Δ cup (119909 119910) isin 119884 times 119884 119909 isin 119883119894 119910 isin 119883

119894+1 119894 = 1 119898 By

(i) it follows that 119891 preserves edges Now let 119891119899119909 rarr 119909

lowast in119884 such that (119891119899119909 119891

119899+1119909) isin 119864(119866) for all 119899 ge 1 then in view of

(33) sequence 119891119899119909 has infinitely many terms in each 119883

119894so

that one can easily extract a subsequence of 119891119899119909 convergingto 119909lowast in each 119883

119894 Since 119883

119894rsquos are closed then 119909

lowastisin ⋂119898

119894=1119883119894

Now it is easy to form a subsequence 119891119899119896119909 in some 119883

119895

119895 isin 1 119898 such that (119891119899119896119909 119909lowast) isin 119864(119866) for 119896 ge 1 and

it indicates that 119866 is weakly connected (119862119891)-graph and thus

conclusion follows fromTheorem 18

Remark 37 (see [31Theorem 21(1)]) It can be deduced fromTheorem 36 if (119883 119889) is a metric space

On the same lines as in proof of Theorem 36 we obtainthe following consequence of Theorem 28

Theorem38 Let (119883 119889) be a complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

and let 119891 119884 rarr 119884 Further suppose that

(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) there exist 120572 120573 120574 ge 0with 119904120572+119904(119904+1)120573+119904(119904+1)120574 lt 1

such that

119889 (119891119909 119891119910)

le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(34)

whenever 119909 isin 119883119894 119910 isin 119883

119894+1 where 119883

119898+1= 1198831

Then 119891 has a fixed point 119905 isin ⋂119898

119894=1119883119894

Remark 39 [32 Theorem 7] and [33 Theorem 31] can bededuced from Theorem 38 but it does not ensure 119891 to be aPicard operator

Remark 40 We note that in proof [32 Theorem 7] theauthorrsquos argument to prove that is 119866 (as assumed in proofof Theorem 36) a (119862)-graph is valid only if the terms ofsequence 119909

119899 are Picard iterations otherwise it is void For

example let 119884 = ⋃3

119894=1119883119894where 119883

1= 1119899 119899 is odd cup 0

1198832= 1119899 119899 is even cup 0 119883

3= 0 and define 119891 119884 rarr 119884

as 119891119909 = 1199092 119909 isin 119884 1198832and 119891119909 = 0 119909 isin 119883

2 We see that (33)

is satisfied and 1119899 rarr 0 but 1198833does not contain infinitely

many terms of 119909119899 In the following we give the corrected

argument to prove that 119866 is a (119862)-graph

Let 119909119899

rarr 119909 in 119883 such that (119909119899 119909119899+1

) isin 119864(119866) for all119899 ge 1 Keeping in mind construction of119866 there exists at leastone pair of closed sets 119883

119895 119883119895+1

for some 119895 isin 1 2 119898

such that both sets contain infinitely many terms of sequence119909119899 since 119883

119894rsquos are closed so that 119909 isin 119883

119895cap 119883119895+1

for some119895 isin 1 119898 and thus one can easily extract a subsequencesuch that (119909

119899119896 119909) isin 119864(119866) holds for 119896 ge 1

Now we establish an existence theorem for the solutionof an integral equation as a consequence of our Theorem 18

Theorem 41 Consider the integral equation

119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (35)

where 119896 [119886 119887] times [119886 119887] times R119899 rarr R119899 and 119892 [119886 119887] rarr R119899 iscontinuous Assume that

(i) 119896(119905 119904 sdot) R119899 rarr R119899 is nondecreasing for each 119905 119904 isin

[119886 119887](ii) there exists a (119888)-comparison function 120593 R+ rarr R+

and a continuous function 119901 [119886 119887] times [119886 119887] rarr R+

8 Abstract and Applied Analysis

such that |119896(119905 119904 119909(119904)) minus 119896(119905 119904 119910(119904))| le 119901(119905 119904)120593(|119909(119904) minus

119910(119904)|) for each 119905 119904 isin [119886 119887] and 119909 le 119910 (ie 119909(119905) le 119910(119905)

for all 119905 isin [119886 119887])

(iii) sup119905isin[119886119887]

int119887

119886119901(119905 119904)119889119904 le 1

(iv) there exists 1199090

isin 119862([119886 119887]R119899) such that 1199090(119905) le

int119887

119886119896(119905 119904 119909

0(119904))119889119904 + 119892(119905) for all 119905 isin [119886 119887]

Then the integral equation (35) has a unique solution in theset 119909 isin 119862([119886 119887]R119899) 119909(119905) le 119909

0(119905) or 119909(119905) ge 119909

0(119905) for all

119905 isin [119886 119887]

Proof Let (119883 sdot infin

) where 119883 = 119862([119886 119887]R119899) and define amapping 119879 119862([119886 119887]R119899) rarr 119862([119886 119887]R119899) by

119879119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (36)

Consider a graph 119866 consisting of 119881(119866) = 119883 and 119864(119866) =

(119909 119910) isin 119883 times 119883 119909(119905) le 119910(119905) for all 119905 isin [119886 119887] From property(i) we observe that the mapping 119879 is nondecreasing thus 119879

preserves edges Furthermore 119866 is a (119862)-graph that is forevery nondecreasing sequence 119909

119899 sub 119883 which converges to

some 119911 isin 119883 then 119909119899(119905) le 119911(119905) for all 119905 isin [119886 119887] Now for every

119909 119910 isin 119883 with (119909 119910) isin 119864(119866) we have

1003816100381610038161003816119879119909 (119905) minus 119879119910 (119905)1003816100381610038161003816

le int

119887

119886

1003816100381610038161003816119896 (119905 119904 119909 (119904)) minus 119896 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904

le int

119887

119886

119901 (119905 119904) 120593 (1003816100381610038161003816119909 (119904) minus 119910 (119904)

1003816100381610038161003816) 119889119904

le 120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817infin) int

119887

119886

119901 (119905 119904) 119889119904

(37)

Hence 119889(119879119909 119879119910) le 120593(119889(119909 119910)) From (iv) we have(1199090 1198791199090) isin 119864(119866) so that [119909

0]119866

= 119909 isin 119862([119886 119887]R119899) 119909(119905) le

1199090(119905) or 119909(119905) ge 119909

0(119905) for all 119905 isin [119886 119887] The conclusion follows

fromTheorem 18

Note that Theorem 41 specifies region of solution whichinvokes the novelty of our result

References

[1] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacesrdquo Journal of Functional Analysis vol 30 pp 26ndash37 1989

[2] J Heinonen Lectures on Analysis on Metric Spaces SpringerNew York NY USA 2001

[3] N Bourbaki Topologie Generale Herman Paris France 1974[4] S Czerwik ldquoNonlinear set-valued contraction mappings in

119887-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[5] M Bota A Molnar and C Varga ldquoOn Ekelandrsquos variationalprinciple in 119887-metric spacesrdquo Fixed Point Theory vol 12 no 1pp 21ndash28 2011

[6] M F Bota-Boriceanu andA Petrusel ldquoUlam-Hyers stability foroperatorial equationsrdquo Analele Stiintifice ale Universitatii ldquoAl ICuzardquo din Iasi vol 57 supplement 1 pp 65ndash74 2011

[7] M Pacurar ldquoA fixed point result for 120593-contractions on 119887-metricspaces without the boundedness assumptionrdquo PolytechnicaPosnaniensis InstitutumMathematicum Fasciculi Mathematicino 43 pp 127ndash137 2010

[8] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis Babes-Bolyai vol 41 no 4pp 23ndash27 1996

[9] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[10] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[11] A Petrusel and I A Rus ldquoFixed point theorems in ordered l-spacesrdquo Proceedings of the American Mathematical Society vol134 no 2 pp 411ndash418 2006

[12] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 p 68 1975

[13] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[14] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008

[15] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[16] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997

[17] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquoSeminar on Fixed Point Theory vol 3 pp 3ndash9 1993

[18] T P Petru and M Boriceanu ldquoFixed point results for gener-alized 120593-contraction on a set with two metricsrdquo TopologicalMethods in Nonlinear Analysis vol 33 no 2 pp 315ndash326 2009

[19] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 11 pages 2012

[20] A Amini-Harandi ldquoCoupled and tripled fixed point theory inpartially ordered metric spaces with application to initial valueproblemrdquoMathematical and Computer Modelling vol 57 no 9-10 pp 2343ndash2348 2012

[21] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[22] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012

[23] M Samreen and T Kamran ldquoFixed point theorems for integralG-contractionsrdquo Fixed Point Theory and Applications vol 2013article 149 2013

[24] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[25] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrix

Abstract and Applied Analysis 9

equationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[26] G Gwozdz-Łukawska and J Jachymski ldquoIFS on a metric spacewith a graph structure and extensions of the Kelisky-Rivlintheoremrdquo Journal of Mathematical Analysis and Applicationsvol 356 no 2 pp 453ndash463 2009

[27] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973

[28] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968

[29] S K Chatterjea ldquoFixed-point theoremsrdquo Doklady BolgarskoıAkademii Nauk vol 25 pp 727ndash730 1972

[30] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003

[31] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010

[32] F Bojor ldquoFixed point theorems for Reich type contractionson metric spaces with a graphrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 9 pp 3895ndash3901 2012

[33] M A Petric ldquoSome remarks concerning Ciric-Reich-Rus oper-atorsrdquo Creative Mathematics and Informatics vol 18 no 2 pp188ndash193 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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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

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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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

Abstract and Applied Analysis 5

Slightly strengthening the continuity condition on 119891 ournext theorem deals with the graph 119866 which may fail to havethe property that there is 119911

0in119883 for which (119911

0 1198911199110) is an edge

in 119866

Theorem22 Let (119883 119889) be a complete 119887-metric space and let119891be a 119887-(120593 119866) contraction where 120593 is a 119887-comparison functionAssume that 119889 is continuous 119891 is orbitally continuous andthere is 119911

0in119883 for which 119891119911

0isin [1199110]119866 Then for any 119910 isin [119911

0]119866

119891119899119910 rarr 119905 isin 119883 where 119905 is a fixed point of 119891

Proof It follows from Proposition 13 that 1198911198991199110 is a Cauchy

sequence in 119883 Since 119883 is complete there exists 119905 isin 119883

such that 1198911198991199110

rarr 119905 Since 119891 is orbitally continuous thenlim119899rarrinfin

1198911198911198991199110

= 119891119905 which yields 119891119905 = 119905 Let 119910 isin [1199110]119866be

arbitrary then from Lemma 12 lim119899rarrinfin

119891119899119910 = 119905

Remark 23 In addition to the hypothesis ofTheorems 21 and22 if we assume that 119866 is weakly connected then 119891 willbecome Picard operator [11] on 119883

Remark 24 Theorem 18 generalizesextends claims 40 and

50 of [10 Theorem 32] and [7 Theorem 4(1)] Theorem 21generalizes claims 20 and 3

0 of [10Theorem 33]Theorem 22generalizes claims 2

0 and 30 of [10 Theorem 34] and thus

generalizes extends results ofNieto andRodrıguez-Lopez [24Theorems 21 and 23] Petrusel andRus [11Theorem43] andRan and Reurings [25 Theorem 21] We mention here thatTheorem 18 can not be improved using comparison functioninstead of 119887-comparison function (see Gwozdz-Łukawskaand Jachymski [26 Example 2])

We observe that Theorem 22 can be used to extendfamous fixed point theorem of Edelstein to the case of119887-metric space We need to define notion of 120598-chainableproperty for 119887-metric space

Definition 25 A 119887-metric space (119883 119889) is said to be 120598-chainable for some 120598 gt 0 if for119909 119910 isin 119883 there exist119909

119894isin 119883 119894 =

0 1 2 119897 with 1199090

= 119909 119909119897= 119910 such that 119889(119909

119894minus1 119909119894) lt 120598 for

119894 = 1 2 119897

Corollary 26 Let (119883 119889) be a complete 120598-chainable 119887-metricspace Assume that 119889 is continuous and there exists a 119887-comparison function 120593 R+ rarr R+ such that 119891 119883 rarr 119883

satisfying

119889 (119909 119910) lt 120598 119894119898119901119897119894119890119904 119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) (16)

for all 119909 119910 isin 119883 Then 119891 is a Picard operator

Proof Consider a graph 119866 consisting of 119881(119866) = 119883 and(119909 119910) isin 119864(119866) if and only if 119889(119909 119910) lt 120598 Since119883 is 120598-chainable119866 is weakly connected Let (119909 119910) isin 119864(119866) and from (16) wehave

119889 (119891119909 119891119910) le 120593 (119889 (119909 119910)) lt 119889 (119909 119910) lt 120598 (17)

Then 119889(119891119909 119891119910) isin 119864(119866) Therefore in view of (16) 119891 is 119887-(120593 119866) contraction Further (16) implies that 119891 is continuousNow the conclusion follows by usingTheorem 22

Now we establish a fixed point theorem using a generalcontractive condition

Theorem 27 Let (119883 119889) be a complete 119887-metric space and let119866 be a (119862

119891)-graph in119883times119883 such that119881(119866) = 119883 and 119891 119883 rarr

119883 is an edge-preserving mapping Assume that 119889 is continuousand there exist 120572 120573 120574 ge 0 with 119904120572 + (119904 + 1)120573 + 119904(119904 + 1)120574 lt 1

and for all (119909 119910) isin 119864(119866)

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(18)

If there is 1199110in119883 for which (119911

0 1198911199110) is an edge in119866 then119891 has

a fixed point in [1199110]119866

Proof Since 119891 is edge-preserving then (1198911198991199110 119891119899+1

1199110) isin

119864(119866) for all 119899 isin N From (18) and using (d3) it follows that

119889 (119891119899

1199110 119891119899+1

1199110)

le 120572119889 (119891119899minus1

1199110 119891119899

1199110)

+ 120573 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1z0)]

+ 120574119904 [119889 (119891119899minus1

1199110 119891119899

1199110) + 119889 (119891

119899

1199110 119891119899+1

1199110)]

(19)

On rearranging

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904] 119889 (119891

119899minus1

1199110 119891119899

1199110) (20)

Repeating iteratively we have

119889 (119891119899

1199110 119891119899+1

1199110) le [

120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119899

119889 (1199110 1198911199110) (21)

For 119898 gt 119899 ge 1 and using (d3) Definition 1 we have

119889 (119891119899

1199110 119891119898

1199110)

le 119904119889 (119891119899

1199110 119891119899+1

1199110) + 1199042

119889 (119891119899+1

1199110 119891119899+2

1199110)

+ sdot sdot sdot + 119904119898minus119899

119889 (119891119898minus1

1199110 119891119898

1199110)

le1

119904119899minus1119889 (1199110 1198911199110)

119898minus1

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(using (21))

lt1

119904119899minus1119889 (1199110 1198911199110)

infin

sum

119895=119899

119904119895

[120572 + 120573 + 120574119904

1 minus 120573 minus 120574119904]

119895

(22)

Since 119904[(120572 + 120573 + 120574119904)(1 minus 120573 minus 120574119904)] lt 1 then 1198911198991199110 is a

Cauchy sequence in 119883 By completeness of 119883 the sequence1198911198991199110 converges to 119905 isin 119883 Since 119866 is a (119862

119891)-graph there

exists a subsequence 1198911198991198961199110 and anatural number119901 such that

6 Abstract and Applied Analysis

(1198911198991198961199110 119905) isin 119864(119866) for all 119896 ge 119901 Using (18) for all 119896 ge 119901 we

have

119889 (119891119899119896+11199110 119891119905)

le 120572119889 (1198911198991198961199110 119905) + 120573 [119889 (119891

1198991198961199110 119891119899119896+11199110) + 119889 (119905 119891119905)]

+ 120574 [119889 (1198911198991198961199110 119891119905) + 119889 (119905 119891

119899119896+11199110)]

(23)

Since the 119887-metric 119889 is continuous and 120573 + 120574 lt 1 letting119896 rarr infin inequality (23) yields 119891119905 = 119905 Also note that(1199110 1198911199110 11989121199110 119891

11989911199110 119891

119899119901 119905) is a path in 119866 and hencein 119866 therefore 119905 isin [119911

0]119866

We note thatTheorem 27 does not guarantee the unique-ness of fixed point but this can be accomplished under someassumptions as in the following theorem

Theorem 28 In addition to the hypothesis ofTheorem 27 onefurther assumes that if 120572 + 2119904120573 + 2120574 lt 1 for the same set of120572 120573 120574 ge 0 and for any two fixed points 119905

1 1199052exist119911 isin 119883 such that

(1199051 119911) and (119905

2 119911) isin 119864(119866) Then 119891 has a unique fixed point

Proof Let 1199051and 1199052be two fixed points of 119891 and then there

exists 119911 isin 119883 such that (1199051 119911) (119905

2 119911) isin 119864(119866) By induction we

have (1199051 119891119899119911) (1199052 119891119899119911) isin 119864(119866) for all 119899 = 0 1 From (18)

we have

119889 (1199051 119891119899

119911) le 120572119889 (1199051 119891119899minus1

119911) + 120573119889 (119891119899minus1

119911 119891119899

119911)

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

le 120572119889 (1199051 119891119899minus1

119911) + 120573119904 [119889 (119891119899minus1

119911 1199051) + 119889 (119905

1 119891119899

119911)]

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

(24)

On rearranging

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574] 119889 (1199051 119891119899minus1

119911) forall119899 = 1 2

(25)

Continuing recursively (25) gives

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574]

119899

119889 (1199051 119911) (26)

Since [(120572+119904120573+120574)(1minus119904120573minus120574)] lt 1 then lim119899rarrinfin

119889(1199051 119891119899119911) =

0 Similarly one can show that lim119899rarrinfin

119889(1199052 119891119899119911) = 0 which

by using (d3) Definition 1 infers that 119889(1199051 1199052) = 0

Suppose that (119883 ⪯) is a partially ordered set Considergraph 119866

2consisting of 119864(119866

2) = (119909 119910) isin 119883 times 119883 119909 ⪯

119910 or119910 ⪯ 119909 and 119881(1198662) coincides with 119883 We note that

if a self-mapping 119891 is monotone with respect to the orderthen for graph 119866

2 it is obvious that 119891 is edge-preserving

or equivalently we can say that 119891 maps comparable elementsonto comparable elements

Following corollaries are the direct consequences ofTheorem 28

Corollary 29 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists 120572 120573 120574 ge 0 with 120572 + 2120573 + 2120574 lt 1 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(27)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 30 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (28)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 31 Let (119883 119889) be a complete metric space where 119883

is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (29)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Remark 32 Wenote that inTheorem 27 the condition ldquothereis 1199110in 119883 for which (119911

0 1198911199110) is an edge in 119866rdquo yields 119891

1198991199110

rarr

119905 where 119905 isin 119883 is a fixed point of 119891 Consider a graph119866 = (119883119883 times 119883) For such graph under the assumptions ofTheorems 27 and 28 it infers that119891 is a Picard operatorThusmany standard fixed point theorems can be easily deducedfromTheorem 28 as follows

Abstract and Applied Analysis 7

Corollary 33 (Hardy and Rogers [27]) Let (119883 119889) be acomplete metric space and let 119891 119883 rarr 119883 Suppose that thereexists constants 120572 120573 120574 ge 0 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(30)

for all 119909 119910 isin 119883 where 120572 + 2120573 + 2120574 lt 1 then 119891 has a uniquefixed point in 119883

Corollary 34 (Kannan [28]) Let (119883 119889) be a complete metricspace and let119891 119883 rarr 119883 Suppose that there exists a constants119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (31)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

Corollary 35 (Chatterjea [29]) Let (119883 119889) be a completemetric space and let 119891 119883 rarr 119883 Suppose that there existsa constants 119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (32)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

3 Applications

Let 119883 be a nonempty set let 119898 be a positive integer 119883119894119898

119894=1

be nonempty closed subsets of 119883 and let 119891 ⋃119898

119894=1119883119894

rarr

⋃119898

119894=1119883119894be an operatorThen119883 = ⋃

119898

119894=1119883119894is known as cyclic

representation of 119883 wrt 119891 if

119891 (1198831) sub 119883

2 119891 (119883

119898minus1) sub 119883

119898 119891 (119883

119898) sub 119883

1

(33)

and operator 119891 is known as cyclic operator [30]

Theorem 36 Let (119883 119889) be complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

120593 R+ rarr R+ be a 119887-comparison function and let 119891 119884 rarr

119884 Further suppose that(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) 119889(119891119909 119891119910) le 120593(119889(119909 119910)) whenever 119909 isin 119883119894 119910 isin 119883

119894+1

where 119883119898+1

= 1198831

Then 119891 has a unique fixed point 119905 isin ⋂119898

119894=1119883119894and 119891

119899119910 rarr 119905 for

any 119910 isin ⋃119898

119894=1119883119894

Proof We note that (119884 119889) is complete 119887-metric space Let usconsider a graph 119866 consisting of 119881(119866) = 119884 and 119864(119866) =

Δ cup (119909 119910) isin 119884 times 119884 119909 isin 119883119894 119910 isin 119883

119894+1 119894 = 1 119898 By

(i) it follows that 119891 preserves edges Now let 119891119899119909 rarr 119909

lowast in119884 such that (119891119899119909 119891

119899+1119909) isin 119864(119866) for all 119899 ge 1 then in view of

(33) sequence 119891119899119909 has infinitely many terms in each 119883

119894so

that one can easily extract a subsequence of 119891119899119909 convergingto 119909lowast in each 119883

119894 Since 119883

119894rsquos are closed then 119909

lowastisin ⋂119898

119894=1119883119894

Now it is easy to form a subsequence 119891119899119896119909 in some 119883

119895

119895 isin 1 119898 such that (119891119899119896119909 119909lowast) isin 119864(119866) for 119896 ge 1 and

it indicates that 119866 is weakly connected (119862119891)-graph and thus

conclusion follows fromTheorem 18

Remark 37 (see [31Theorem 21(1)]) It can be deduced fromTheorem 36 if (119883 119889) is a metric space

On the same lines as in proof of Theorem 36 we obtainthe following consequence of Theorem 28

Theorem38 Let (119883 119889) be a complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

and let 119891 119884 rarr 119884 Further suppose that

(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) there exist 120572 120573 120574 ge 0with 119904120572+119904(119904+1)120573+119904(119904+1)120574 lt 1

such that

119889 (119891119909 119891119910)

le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(34)

whenever 119909 isin 119883119894 119910 isin 119883

119894+1 where 119883

119898+1= 1198831

Then 119891 has a fixed point 119905 isin ⋂119898

119894=1119883119894

Remark 39 [32 Theorem 7] and [33 Theorem 31] can bededuced from Theorem 38 but it does not ensure 119891 to be aPicard operator

Remark 40 We note that in proof [32 Theorem 7] theauthorrsquos argument to prove that is 119866 (as assumed in proofof Theorem 36) a (119862)-graph is valid only if the terms ofsequence 119909

119899 are Picard iterations otherwise it is void For

example let 119884 = ⋃3

119894=1119883119894where 119883

1= 1119899 119899 is odd cup 0

1198832= 1119899 119899 is even cup 0 119883

3= 0 and define 119891 119884 rarr 119884

as 119891119909 = 1199092 119909 isin 119884 1198832and 119891119909 = 0 119909 isin 119883

2 We see that (33)

is satisfied and 1119899 rarr 0 but 1198833does not contain infinitely

many terms of 119909119899 In the following we give the corrected

argument to prove that 119866 is a (119862)-graph

Let 119909119899

rarr 119909 in 119883 such that (119909119899 119909119899+1

) isin 119864(119866) for all119899 ge 1 Keeping in mind construction of119866 there exists at leastone pair of closed sets 119883

119895 119883119895+1

for some 119895 isin 1 2 119898

such that both sets contain infinitely many terms of sequence119909119899 since 119883

119894rsquos are closed so that 119909 isin 119883

119895cap 119883119895+1

for some119895 isin 1 119898 and thus one can easily extract a subsequencesuch that (119909

119899119896 119909) isin 119864(119866) holds for 119896 ge 1

Now we establish an existence theorem for the solutionof an integral equation as a consequence of our Theorem 18

Theorem 41 Consider the integral equation

119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (35)

where 119896 [119886 119887] times [119886 119887] times R119899 rarr R119899 and 119892 [119886 119887] rarr R119899 iscontinuous Assume that

(i) 119896(119905 119904 sdot) R119899 rarr R119899 is nondecreasing for each 119905 119904 isin

[119886 119887](ii) there exists a (119888)-comparison function 120593 R+ rarr R+

and a continuous function 119901 [119886 119887] times [119886 119887] rarr R+

8 Abstract and Applied Analysis

such that |119896(119905 119904 119909(119904)) minus 119896(119905 119904 119910(119904))| le 119901(119905 119904)120593(|119909(119904) minus

119910(119904)|) for each 119905 119904 isin [119886 119887] and 119909 le 119910 (ie 119909(119905) le 119910(119905)

for all 119905 isin [119886 119887])

(iii) sup119905isin[119886119887]

int119887

119886119901(119905 119904)119889119904 le 1

(iv) there exists 1199090

isin 119862([119886 119887]R119899) such that 1199090(119905) le

int119887

119886119896(119905 119904 119909

0(119904))119889119904 + 119892(119905) for all 119905 isin [119886 119887]

Then the integral equation (35) has a unique solution in theset 119909 isin 119862([119886 119887]R119899) 119909(119905) le 119909

0(119905) or 119909(119905) ge 119909

0(119905) for all

119905 isin [119886 119887]

Proof Let (119883 sdot infin

) where 119883 = 119862([119886 119887]R119899) and define amapping 119879 119862([119886 119887]R119899) rarr 119862([119886 119887]R119899) by

119879119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (36)

Consider a graph 119866 consisting of 119881(119866) = 119883 and 119864(119866) =

(119909 119910) isin 119883 times 119883 119909(119905) le 119910(119905) for all 119905 isin [119886 119887] From property(i) we observe that the mapping 119879 is nondecreasing thus 119879

preserves edges Furthermore 119866 is a (119862)-graph that is forevery nondecreasing sequence 119909

119899 sub 119883 which converges to

some 119911 isin 119883 then 119909119899(119905) le 119911(119905) for all 119905 isin [119886 119887] Now for every

119909 119910 isin 119883 with (119909 119910) isin 119864(119866) we have

1003816100381610038161003816119879119909 (119905) minus 119879119910 (119905)1003816100381610038161003816

le int

119887

119886

1003816100381610038161003816119896 (119905 119904 119909 (119904)) minus 119896 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904

le int

119887

119886

119901 (119905 119904) 120593 (1003816100381610038161003816119909 (119904) minus 119910 (119904)

1003816100381610038161003816) 119889119904

le 120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817infin) int

119887

119886

119901 (119905 119904) 119889119904

(37)

Hence 119889(119879119909 119879119910) le 120593(119889(119909 119910)) From (iv) we have(1199090 1198791199090) isin 119864(119866) so that [119909

0]119866

= 119909 isin 119862([119886 119887]R119899) 119909(119905) le

1199090(119905) or 119909(119905) ge 119909

0(119905) for all 119905 isin [119886 119887] The conclusion follows

fromTheorem 18

Note that Theorem 41 specifies region of solution whichinvokes the novelty of our result

References

[1] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacesrdquo Journal of Functional Analysis vol 30 pp 26ndash37 1989

[2] J Heinonen Lectures on Analysis on Metric Spaces SpringerNew York NY USA 2001

[3] N Bourbaki Topologie Generale Herman Paris France 1974[4] S Czerwik ldquoNonlinear set-valued contraction mappings in

119887-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[5] M Bota A Molnar and C Varga ldquoOn Ekelandrsquos variationalprinciple in 119887-metric spacesrdquo Fixed Point Theory vol 12 no 1pp 21ndash28 2011

[6] M F Bota-Boriceanu andA Petrusel ldquoUlam-Hyers stability foroperatorial equationsrdquo Analele Stiintifice ale Universitatii ldquoAl ICuzardquo din Iasi vol 57 supplement 1 pp 65ndash74 2011

[7] M Pacurar ldquoA fixed point result for 120593-contractions on 119887-metricspaces without the boundedness assumptionrdquo PolytechnicaPosnaniensis InstitutumMathematicum Fasciculi Mathematicino 43 pp 127ndash137 2010

[8] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis Babes-Bolyai vol 41 no 4pp 23ndash27 1996

[9] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[10] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[11] A Petrusel and I A Rus ldquoFixed point theorems in ordered l-spacesrdquo Proceedings of the American Mathematical Society vol134 no 2 pp 411ndash418 2006

[12] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 p 68 1975

[13] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[14] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008

[15] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[16] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997

[17] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquoSeminar on Fixed Point Theory vol 3 pp 3ndash9 1993

[18] T P Petru and M Boriceanu ldquoFixed point results for gener-alized 120593-contraction on a set with two metricsrdquo TopologicalMethods in Nonlinear Analysis vol 33 no 2 pp 315ndash326 2009

[19] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 11 pages 2012

[20] A Amini-Harandi ldquoCoupled and tripled fixed point theory inpartially ordered metric spaces with application to initial valueproblemrdquoMathematical and Computer Modelling vol 57 no 9-10 pp 2343ndash2348 2012

[21] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[22] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012

[23] M Samreen and T Kamran ldquoFixed point theorems for integralG-contractionsrdquo Fixed Point Theory and Applications vol 2013article 149 2013

[24] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[25] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrix

Abstract and Applied Analysis 9

equationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[26] G Gwozdz-Łukawska and J Jachymski ldquoIFS on a metric spacewith a graph structure and extensions of the Kelisky-Rivlintheoremrdquo Journal of Mathematical Analysis and Applicationsvol 356 no 2 pp 453ndash463 2009

[27] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973

[28] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968

[29] S K Chatterjea ldquoFixed-point theoremsrdquo Doklady BolgarskoıAkademii Nauk vol 25 pp 727ndash730 1972

[30] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003

[31] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010

[32] F Bojor ldquoFixed point theorems for Reich type contractionson metric spaces with a graphrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 9 pp 3895ndash3901 2012

[33] M A Petric ldquoSome remarks concerning Ciric-Reich-Rus oper-atorsrdquo Creative Mathematics and Informatics vol 18 no 2 pp188ndash193 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

(1198911198991198961199110 119905) isin 119864(119866) for all 119896 ge 119901 Using (18) for all 119896 ge 119901 we

have

119889 (119891119899119896+11199110 119891119905)

le 120572119889 (1198911198991198961199110 119905) + 120573 [119889 (119891

1198991198961199110 119891119899119896+11199110) + 119889 (119905 119891119905)]

+ 120574 [119889 (1198911198991198961199110 119891119905) + 119889 (119905 119891

119899119896+11199110)]

(23)

Since the 119887-metric 119889 is continuous and 120573 + 120574 lt 1 letting119896 rarr infin inequality (23) yields 119891119905 = 119905 Also note that(1199110 1198911199110 11989121199110 119891

11989911199110 119891

119899119901 119905) is a path in 119866 and hencein 119866 therefore 119905 isin [119911

0]119866

We note thatTheorem 27 does not guarantee the unique-ness of fixed point but this can be accomplished under someassumptions as in the following theorem

Theorem 28 In addition to the hypothesis ofTheorem 27 onefurther assumes that if 120572 + 2119904120573 + 2120574 lt 1 for the same set of120572 120573 120574 ge 0 and for any two fixed points 119905

1 1199052exist119911 isin 119883 such that

(1199051 119911) and (119905

2 119911) isin 119864(119866) Then 119891 has a unique fixed point

Proof Let 1199051and 1199052be two fixed points of 119891 and then there

exists 119911 isin 119883 such that (1199051 119911) (119905

2 119911) isin 119864(119866) By induction we

have (1199051 119891119899119911) (1199052 119891119899119911) isin 119864(119866) for all 119899 = 0 1 From (18)

we have

119889 (1199051 119891119899

119911) le 120572119889 (1199051 119891119899minus1

119911) + 120573119889 (119891119899minus1

119911 119891119899

119911)

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

le 120572119889 (1199051 119891119899minus1

119911) + 120573119904 [119889 (119891119899minus1

119911 1199051) + 119889 (119905

1 119891119899

119911)]

+ 120574 [119889 (1199051 119891119899

119911) + 119889 (119891119899minus1

119911 1199051)]

(24)

On rearranging

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574] 119889 (1199051 119891119899minus1

119911) forall119899 = 1 2

(25)

Continuing recursively (25) gives

119889 (1199051 119891119899

119911) le [120572 + 119904120573 + 120574

1 minus 119904120573 minus 120574]

119899

119889 (1199051 119911) (26)

Since [(120572+119904120573+120574)(1minus119904120573minus120574)] lt 1 then lim119899rarrinfin

119889(1199051 119891119899119911) =

0 Similarly one can show that lim119899rarrinfin

119889(1199052 119891119899119911) = 0 which

by using (d3) Definition 1 infers that 119889(1199051 1199052) = 0

Suppose that (119883 ⪯) is a partially ordered set Considergraph 119866

2consisting of 119864(119866

2) = (119909 119910) isin 119883 times 119883 119909 ⪯

119910 or119910 ⪯ 119909 and 119881(1198662) coincides with 119883 We note that

if a self-mapping 119891 is monotone with respect to the orderthen for graph 119866

2 it is obvious that 119891 is edge-preserving

or equivalently we can say that 119891 maps comparable elementsonto comparable elements

Following corollaries are the direct consequences ofTheorem 28

Corollary 29 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists 120572 120573 120574 ge 0 with 120572 + 2120573 + 2120574 lt 1 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(27)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 30 Let (119883 119889) be a complete metric space where119883 is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (28)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Corollary 31 Let (119883 119889) be a complete metric space where 119883

is partially ordered set with respect to ⪯ Let 119891 119883 rarr 119883

be nondecreasing (or nonincreasing) with respect to ⪯ Assumethat there exists a constant 0 lt 119888 lt 12 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (29)

for all comparable 119909 119910 isin 119883 If the following conditions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) for nondecreasing (or nonincreasing) sequence 119909119899 rarr

119909 isin 119883 there exists a subsequence 119909119899119896

such that 119909119899119896

⪯

119909 for all 119896

Then 119891 has a fixed point Moreover if for all 119909 119910 isin 119883 thereexists 119911 isin 119883 such that 119909 ⪯ 119911 and 119910 ⪯ 119911 then the fixed point isunique

Remark 32 Wenote that inTheorem 27 the condition ldquothereis 1199110in 119883 for which (119911

0 1198911199110) is an edge in 119866rdquo yields 119891

1198991199110

rarr

119905 where 119905 isin 119883 is a fixed point of 119891 Consider a graph119866 = (119883119883 times 119883) For such graph under the assumptions ofTheorems 27 and 28 it infers that119891 is a Picard operatorThusmany standard fixed point theorems can be easily deducedfromTheorem 28 as follows

Abstract and Applied Analysis 7

Corollary 33 (Hardy and Rogers [27]) Let (119883 119889) be acomplete metric space and let 119891 119883 rarr 119883 Suppose that thereexists constants 120572 120573 120574 ge 0 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(30)

for all 119909 119910 isin 119883 where 120572 + 2120573 + 2120574 lt 1 then 119891 has a uniquefixed point in 119883

Corollary 34 (Kannan [28]) Let (119883 119889) be a complete metricspace and let119891 119883 rarr 119883 Suppose that there exists a constants119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (31)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

Corollary 35 (Chatterjea [29]) Let (119883 119889) be a completemetric space and let 119891 119883 rarr 119883 Suppose that there existsa constants 119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (32)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

3 Applications

Let 119883 be a nonempty set let 119898 be a positive integer 119883119894119898

119894=1

be nonempty closed subsets of 119883 and let 119891 ⋃119898

119894=1119883119894

rarr

⋃119898

119894=1119883119894be an operatorThen119883 = ⋃

119898

119894=1119883119894is known as cyclic

representation of 119883 wrt 119891 if

119891 (1198831) sub 119883

2 119891 (119883

119898minus1) sub 119883

119898 119891 (119883

119898) sub 119883

1

(33)

and operator 119891 is known as cyclic operator [30]

Theorem 36 Let (119883 119889) be complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

120593 R+ rarr R+ be a 119887-comparison function and let 119891 119884 rarr

119884 Further suppose that(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) 119889(119891119909 119891119910) le 120593(119889(119909 119910)) whenever 119909 isin 119883119894 119910 isin 119883

119894+1

where 119883119898+1

= 1198831

Then 119891 has a unique fixed point 119905 isin ⋂119898

119894=1119883119894and 119891

119899119910 rarr 119905 for

any 119910 isin ⋃119898

119894=1119883119894

Proof We note that (119884 119889) is complete 119887-metric space Let usconsider a graph 119866 consisting of 119881(119866) = 119884 and 119864(119866) =

Δ cup (119909 119910) isin 119884 times 119884 119909 isin 119883119894 119910 isin 119883

119894+1 119894 = 1 119898 By

(i) it follows that 119891 preserves edges Now let 119891119899119909 rarr 119909

lowast in119884 such that (119891119899119909 119891

119899+1119909) isin 119864(119866) for all 119899 ge 1 then in view of

(33) sequence 119891119899119909 has infinitely many terms in each 119883

119894so

that one can easily extract a subsequence of 119891119899119909 convergingto 119909lowast in each 119883

119894 Since 119883

119894rsquos are closed then 119909

lowastisin ⋂119898

119894=1119883119894

Now it is easy to form a subsequence 119891119899119896119909 in some 119883

119895

119895 isin 1 119898 such that (119891119899119896119909 119909lowast) isin 119864(119866) for 119896 ge 1 and

it indicates that 119866 is weakly connected (119862119891)-graph and thus

conclusion follows fromTheorem 18

Remark 37 (see [31Theorem 21(1)]) It can be deduced fromTheorem 36 if (119883 119889) is a metric space

On the same lines as in proof of Theorem 36 we obtainthe following consequence of Theorem 28

Theorem38 Let (119883 119889) be a complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

and let 119891 119884 rarr 119884 Further suppose that

(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) there exist 120572 120573 120574 ge 0with 119904120572+119904(119904+1)120573+119904(119904+1)120574 lt 1

such that

119889 (119891119909 119891119910)

le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(34)

whenever 119909 isin 119883119894 119910 isin 119883

119894+1 where 119883

119898+1= 1198831

Then 119891 has a fixed point 119905 isin ⋂119898

119894=1119883119894

Remark 39 [32 Theorem 7] and [33 Theorem 31] can bededuced from Theorem 38 but it does not ensure 119891 to be aPicard operator

Remark 40 We note that in proof [32 Theorem 7] theauthorrsquos argument to prove that is 119866 (as assumed in proofof Theorem 36) a (119862)-graph is valid only if the terms ofsequence 119909

119899 are Picard iterations otherwise it is void For

example let 119884 = ⋃3

119894=1119883119894where 119883

1= 1119899 119899 is odd cup 0

1198832= 1119899 119899 is even cup 0 119883

3= 0 and define 119891 119884 rarr 119884

as 119891119909 = 1199092 119909 isin 119884 1198832and 119891119909 = 0 119909 isin 119883

2 We see that (33)

is satisfied and 1119899 rarr 0 but 1198833does not contain infinitely

many terms of 119909119899 In the following we give the corrected

argument to prove that 119866 is a (119862)-graph

Let 119909119899

rarr 119909 in 119883 such that (119909119899 119909119899+1

) isin 119864(119866) for all119899 ge 1 Keeping in mind construction of119866 there exists at leastone pair of closed sets 119883

119895 119883119895+1

for some 119895 isin 1 2 119898

such that both sets contain infinitely many terms of sequence119909119899 since 119883

119894rsquos are closed so that 119909 isin 119883

119895cap 119883119895+1

for some119895 isin 1 119898 and thus one can easily extract a subsequencesuch that (119909

119899119896 119909) isin 119864(119866) holds for 119896 ge 1

Now we establish an existence theorem for the solutionof an integral equation as a consequence of our Theorem 18

Theorem 41 Consider the integral equation

119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (35)

where 119896 [119886 119887] times [119886 119887] times R119899 rarr R119899 and 119892 [119886 119887] rarr R119899 iscontinuous Assume that

(i) 119896(119905 119904 sdot) R119899 rarr R119899 is nondecreasing for each 119905 119904 isin

[119886 119887](ii) there exists a (119888)-comparison function 120593 R+ rarr R+

and a continuous function 119901 [119886 119887] times [119886 119887] rarr R+

8 Abstract and Applied Analysis

such that |119896(119905 119904 119909(119904)) minus 119896(119905 119904 119910(119904))| le 119901(119905 119904)120593(|119909(119904) minus

119910(119904)|) for each 119905 119904 isin [119886 119887] and 119909 le 119910 (ie 119909(119905) le 119910(119905)

for all 119905 isin [119886 119887])

(iii) sup119905isin[119886119887]

int119887

119886119901(119905 119904)119889119904 le 1

(iv) there exists 1199090

isin 119862([119886 119887]R119899) such that 1199090(119905) le

int119887

119886119896(119905 119904 119909

0(119904))119889119904 + 119892(119905) for all 119905 isin [119886 119887]

Then the integral equation (35) has a unique solution in theset 119909 isin 119862([119886 119887]R119899) 119909(119905) le 119909

0(119905) or 119909(119905) ge 119909

0(119905) for all

119905 isin [119886 119887]

Proof Let (119883 sdot infin

) where 119883 = 119862([119886 119887]R119899) and define amapping 119879 119862([119886 119887]R119899) rarr 119862([119886 119887]R119899) by

119879119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (36)

Consider a graph 119866 consisting of 119881(119866) = 119883 and 119864(119866) =

(119909 119910) isin 119883 times 119883 119909(119905) le 119910(119905) for all 119905 isin [119886 119887] From property(i) we observe that the mapping 119879 is nondecreasing thus 119879

preserves edges Furthermore 119866 is a (119862)-graph that is forevery nondecreasing sequence 119909

119899 sub 119883 which converges to

some 119911 isin 119883 then 119909119899(119905) le 119911(119905) for all 119905 isin [119886 119887] Now for every

119909 119910 isin 119883 with (119909 119910) isin 119864(119866) we have

1003816100381610038161003816119879119909 (119905) minus 119879119910 (119905)1003816100381610038161003816

le int

119887

119886

1003816100381610038161003816119896 (119905 119904 119909 (119904)) minus 119896 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904

le int

119887

119886

119901 (119905 119904) 120593 (1003816100381610038161003816119909 (119904) minus 119910 (119904)

1003816100381610038161003816) 119889119904

le 120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817infin) int

119887

119886

119901 (119905 119904) 119889119904

(37)

Hence 119889(119879119909 119879119910) le 120593(119889(119909 119910)) From (iv) we have(1199090 1198791199090) isin 119864(119866) so that [119909

0]119866

= 119909 isin 119862([119886 119887]R119899) 119909(119905) le

1199090(119905) or 119909(119905) ge 119909

0(119905) for all 119905 isin [119886 119887] The conclusion follows

fromTheorem 18

Note that Theorem 41 specifies region of solution whichinvokes the novelty of our result

References

[1] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacesrdquo Journal of Functional Analysis vol 30 pp 26ndash37 1989

[2] J Heinonen Lectures on Analysis on Metric Spaces SpringerNew York NY USA 2001

[3] N Bourbaki Topologie Generale Herman Paris France 1974[4] S Czerwik ldquoNonlinear set-valued contraction mappings in

119887-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[5] M Bota A Molnar and C Varga ldquoOn Ekelandrsquos variationalprinciple in 119887-metric spacesrdquo Fixed Point Theory vol 12 no 1pp 21ndash28 2011

[6] M F Bota-Boriceanu andA Petrusel ldquoUlam-Hyers stability foroperatorial equationsrdquo Analele Stiintifice ale Universitatii ldquoAl ICuzardquo din Iasi vol 57 supplement 1 pp 65ndash74 2011

[7] M Pacurar ldquoA fixed point result for 120593-contractions on 119887-metricspaces without the boundedness assumptionrdquo PolytechnicaPosnaniensis InstitutumMathematicum Fasciculi Mathematicino 43 pp 127ndash137 2010

[8] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis Babes-Bolyai vol 41 no 4pp 23ndash27 1996

[9] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[10] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[11] A Petrusel and I A Rus ldquoFixed point theorems in ordered l-spacesrdquo Proceedings of the American Mathematical Society vol134 no 2 pp 411ndash418 2006

[12] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 p 68 1975

[13] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[14] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008

[15] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[16] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997

[17] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquoSeminar on Fixed Point Theory vol 3 pp 3ndash9 1993

[18] T P Petru and M Boriceanu ldquoFixed point results for gener-alized 120593-contraction on a set with two metricsrdquo TopologicalMethods in Nonlinear Analysis vol 33 no 2 pp 315ndash326 2009

[19] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 11 pages 2012

[20] A Amini-Harandi ldquoCoupled and tripled fixed point theory inpartially ordered metric spaces with application to initial valueproblemrdquoMathematical and Computer Modelling vol 57 no 9-10 pp 2343ndash2348 2012

[21] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[22] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012

[23] M Samreen and T Kamran ldquoFixed point theorems for integralG-contractionsrdquo Fixed Point Theory and Applications vol 2013article 149 2013

[24] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[25] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrix

Abstract and Applied Analysis 9

equationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[26] G Gwozdz-Łukawska and J Jachymski ldquoIFS on a metric spacewith a graph structure and extensions of the Kelisky-Rivlintheoremrdquo Journal of Mathematical Analysis and Applicationsvol 356 no 2 pp 453ndash463 2009

[27] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973

[28] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968

[29] S K Chatterjea ldquoFixed-point theoremsrdquo Doklady BolgarskoıAkademii Nauk vol 25 pp 727ndash730 1972

[30] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003

[31] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010

[32] F Bojor ldquoFixed point theorems for Reich type contractionson metric spaces with a graphrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 9 pp 3895ndash3901 2012

[33] M A Petric ldquoSome remarks concerning Ciric-Reich-Rus oper-atorsrdquo Creative Mathematics and Informatics vol 18 no 2 pp188ndash193 2009

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

Abstract and Applied Analysis 7

Corollary 33 (Hardy and Rogers [27]) Let (119883 119889) be acomplete metric space and let 119891 119883 rarr 119883 Suppose that thereexists constants 120572 120573 120574 ge 0 such that

119889 (119891119909 119891119910) le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(30)

for all 119909 119910 isin 119883 where 120572 + 2120573 + 2120574 lt 1 then 119891 has a uniquefixed point in 119883

Corollary 34 (Kannan [28]) Let (119883 119889) be a complete metricspace and let119891 119883 rarr 119883 Suppose that there exists a constants119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119909) + 119889 (119910 119891119910)] (31)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

Corollary 35 (Chatterjea [29]) Let (119883 119889) be a completemetric space and let 119891 119883 rarr 119883 Suppose that there existsa constants 119888 such that

119889 (119891119909 119891119910) le 119888 [119889 (119909 119891119910) + 119889 (119910 119891119909)] (32)

for all 119909 119910 isin 119883 where 0 lt 119888 lt 12 then 119891 is Picard operator

3 Applications

Let 119883 be a nonempty set let 119898 be a positive integer 119883119894119898

119894=1

be nonempty closed subsets of 119883 and let 119891 ⋃119898

119894=1119883119894

rarr

⋃119898

119894=1119883119894be an operatorThen119883 = ⋃

119898

119894=1119883119894is known as cyclic

representation of 119883 wrt 119891 if

119891 (1198831) sub 119883

2 119891 (119883

119898minus1) sub 119883

119898 119891 (119883

119898) sub 119883

1

(33)

and operator 119891 is known as cyclic operator [30]

Theorem 36 Let (119883 119889) be complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

120593 R+ rarr R+ be a 119887-comparison function and let 119891 119884 rarr

119884 Further suppose that(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) 119889(119891119909 119891119910) le 120593(119889(119909 119910)) whenever 119909 isin 119883119894 119910 isin 119883

119894+1

where 119883119898+1

= 1198831

Then 119891 has a unique fixed point 119905 isin ⋂119898

119894=1119883119894and 119891

119899119910 rarr 119905 for

any 119910 isin ⋃119898

119894=1119883119894

Proof We note that (119884 119889) is complete 119887-metric space Let usconsider a graph 119866 consisting of 119881(119866) = 119884 and 119864(119866) =

Δ cup (119909 119910) isin 119884 times 119884 119909 isin 119883119894 119910 isin 119883

119894+1 119894 = 1 119898 By

(i) it follows that 119891 preserves edges Now let 119891119899119909 rarr 119909

lowast in119884 such that (119891119899119909 119891

119899+1119909) isin 119864(119866) for all 119899 ge 1 then in view of

(33) sequence 119891119899119909 has infinitely many terms in each 119883

119894so

that one can easily extract a subsequence of 119891119899119909 convergingto 119909lowast in each 119883

119894 Since 119883

119894rsquos are closed then 119909

lowastisin ⋂119898

119894=1119883119894

Now it is easy to form a subsequence 119891119899119896119909 in some 119883

119895

119895 isin 1 119898 such that (119891119899119896119909 119909lowast) isin 119864(119866) for 119896 ge 1 and

it indicates that 119866 is weakly connected (119862119891)-graph and thus

conclusion follows fromTheorem 18

Remark 37 (see [31Theorem 21(1)]) It can be deduced fromTheorem 36 if (119883 119889) is a metric space

On the same lines as in proof of Theorem 36 we obtainthe following consequence of Theorem 28

Theorem38 Let (119883 119889) be a complete 119887-metric space such that119889 is continuous functional on 119883 times 119883 Let 119898 be positive integerlet 119883

119894119898

119894=1be nonempty closed subsets of 119883 let 119884 = ⋃

119898

119894=1119883119894

and let 119891 119884 rarr 119884 Further suppose that

(i) ⋃119898

119894=1119883119894is cyclic representation of 119884 wrt 119891

(ii) there exist 120572 120573 120574 ge 0with 119904120572+119904(119904+1)120573+119904(119904+1)120574 lt 1

such that

119889 (119891119909 119891119910)

le 120572119889 (119909 119910) + 120573 [119889 (119909 119891119909) + 119889 (119910 119891119910)]

+ 120574 [119889 (119909 119891119910) + 119889 (119910 119891119909)]

(34)

whenever 119909 isin 119883119894 119910 isin 119883

119894+1 where 119883

119898+1= 1198831

Then 119891 has a fixed point 119905 isin ⋂119898

119894=1119883119894

Remark 39 [32 Theorem 7] and [33 Theorem 31] can bededuced from Theorem 38 but it does not ensure 119891 to be aPicard operator

Remark 40 We note that in proof [32 Theorem 7] theauthorrsquos argument to prove that is 119866 (as assumed in proofof Theorem 36) a (119862)-graph is valid only if the terms ofsequence 119909

119899 are Picard iterations otherwise it is void For

example let 119884 = ⋃3

119894=1119883119894where 119883

1= 1119899 119899 is odd cup 0

1198832= 1119899 119899 is even cup 0 119883

3= 0 and define 119891 119884 rarr 119884

as 119891119909 = 1199092 119909 isin 119884 1198832and 119891119909 = 0 119909 isin 119883

2 We see that (33)

is satisfied and 1119899 rarr 0 but 1198833does not contain infinitely

many terms of 119909119899 In the following we give the corrected

argument to prove that 119866 is a (119862)-graph

Let 119909119899

rarr 119909 in 119883 such that (119909119899 119909119899+1

) isin 119864(119866) for all119899 ge 1 Keeping in mind construction of119866 there exists at leastone pair of closed sets 119883

119895 119883119895+1

for some 119895 isin 1 2 119898

such that both sets contain infinitely many terms of sequence119909119899 since 119883

119894rsquos are closed so that 119909 isin 119883

119895cap 119883119895+1

for some119895 isin 1 119898 and thus one can easily extract a subsequencesuch that (119909

119899119896 119909) isin 119864(119866) holds for 119896 ge 1

Now we establish an existence theorem for the solutionof an integral equation as a consequence of our Theorem 18

Theorem 41 Consider the integral equation

119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (35)

where 119896 [119886 119887] times [119886 119887] times R119899 rarr R119899 and 119892 [119886 119887] rarr R119899 iscontinuous Assume that

(i) 119896(119905 119904 sdot) R119899 rarr R119899 is nondecreasing for each 119905 119904 isin

[119886 119887](ii) there exists a (119888)-comparison function 120593 R+ rarr R+

and a continuous function 119901 [119886 119887] times [119886 119887] rarr R+

8 Abstract and Applied Analysis

such that |119896(119905 119904 119909(119904)) minus 119896(119905 119904 119910(119904))| le 119901(119905 119904)120593(|119909(119904) minus

119910(119904)|) for each 119905 119904 isin [119886 119887] and 119909 le 119910 (ie 119909(119905) le 119910(119905)

for all 119905 isin [119886 119887])

(iii) sup119905isin[119886119887]

int119887

119886119901(119905 119904)119889119904 le 1

(iv) there exists 1199090

isin 119862([119886 119887]R119899) such that 1199090(119905) le

int119887

119886119896(119905 119904 119909

0(119904))119889119904 + 119892(119905) for all 119905 isin [119886 119887]

Then the integral equation (35) has a unique solution in theset 119909 isin 119862([119886 119887]R119899) 119909(119905) le 119909

0(119905) or 119909(119905) ge 119909

0(119905) for all

119905 isin [119886 119887]

Proof Let (119883 sdot infin

) where 119883 = 119862([119886 119887]R119899) and define amapping 119879 119862([119886 119887]R119899) rarr 119862([119886 119887]R119899) by

119879119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (36)

Consider a graph 119866 consisting of 119881(119866) = 119883 and 119864(119866) =

(119909 119910) isin 119883 times 119883 119909(119905) le 119910(119905) for all 119905 isin [119886 119887] From property(i) we observe that the mapping 119879 is nondecreasing thus 119879

preserves edges Furthermore 119866 is a (119862)-graph that is forevery nondecreasing sequence 119909

119899 sub 119883 which converges to

some 119911 isin 119883 then 119909119899(119905) le 119911(119905) for all 119905 isin [119886 119887] Now for every

119909 119910 isin 119883 with (119909 119910) isin 119864(119866) we have

1003816100381610038161003816119879119909 (119905) minus 119879119910 (119905)1003816100381610038161003816

le int

119887

119886

1003816100381610038161003816119896 (119905 119904 119909 (119904)) minus 119896 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904

le int

119887

119886

119901 (119905 119904) 120593 (1003816100381610038161003816119909 (119904) minus 119910 (119904)

1003816100381610038161003816) 119889119904

le 120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817infin) int

119887

119886

119901 (119905 119904) 119889119904

(37)

Hence 119889(119879119909 119879119910) le 120593(119889(119909 119910)) From (iv) we have(1199090 1198791199090) isin 119864(119866) so that [119909

0]119866

= 119909 isin 119862([119886 119887]R119899) 119909(119905) le

1199090(119905) or 119909(119905) ge 119909

0(119905) for all 119905 isin [119886 119887] The conclusion follows

fromTheorem 18

Note that Theorem 41 specifies region of solution whichinvokes the novelty of our result

References

[1] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacesrdquo Journal of Functional Analysis vol 30 pp 26ndash37 1989

[2] J Heinonen Lectures on Analysis on Metric Spaces SpringerNew York NY USA 2001

[3] N Bourbaki Topologie Generale Herman Paris France 1974[4] S Czerwik ldquoNonlinear set-valued contraction mappings in

119887-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[5] M Bota A Molnar and C Varga ldquoOn Ekelandrsquos variationalprinciple in 119887-metric spacesrdquo Fixed Point Theory vol 12 no 1pp 21ndash28 2011

[6] M F Bota-Boriceanu andA Petrusel ldquoUlam-Hyers stability foroperatorial equationsrdquo Analele Stiintifice ale Universitatii ldquoAl ICuzardquo din Iasi vol 57 supplement 1 pp 65ndash74 2011

[7] M Pacurar ldquoA fixed point result for 120593-contractions on 119887-metricspaces without the boundedness assumptionrdquo PolytechnicaPosnaniensis InstitutumMathematicum Fasciculi Mathematicino 43 pp 127ndash137 2010

[8] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis Babes-Bolyai vol 41 no 4pp 23ndash27 1996

[9] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[10] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[11] A Petrusel and I A Rus ldquoFixed point theorems in ordered l-spacesrdquo Proceedings of the American Mathematical Society vol134 no 2 pp 411ndash418 2006

[12] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 p 68 1975

[13] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[14] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008

[15] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[16] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997

[17] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquoSeminar on Fixed Point Theory vol 3 pp 3ndash9 1993

[18] T P Petru and M Boriceanu ldquoFixed point results for gener-alized 120593-contraction on a set with two metricsrdquo TopologicalMethods in Nonlinear Analysis vol 33 no 2 pp 315ndash326 2009

[19] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 11 pages 2012

[20] A Amini-Harandi ldquoCoupled and tripled fixed point theory inpartially ordered metric spaces with application to initial valueproblemrdquoMathematical and Computer Modelling vol 57 no 9-10 pp 2343ndash2348 2012

[21] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[22] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012

[23] M Samreen and T Kamran ldquoFixed point theorems for integralG-contractionsrdquo Fixed Point Theory and Applications vol 2013article 149 2013

[24] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[25] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrix

Abstract and Applied Analysis 9

equationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[26] G Gwozdz-Łukawska and J Jachymski ldquoIFS on a metric spacewith a graph structure and extensions of the Kelisky-Rivlintheoremrdquo Journal of Mathematical Analysis and Applicationsvol 356 no 2 pp 453ndash463 2009

[27] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973

[28] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968

[29] S K Chatterjea ldquoFixed-point theoremsrdquo Doklady BolgarskoıAkademii Nauk vol 25 pp 727ndash730 1972

[30] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003

[31] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010

[32] F Bojor ldquoFixed point theorems for Reich type contractionson metric spaces with a graphrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 9 pp 3895ndash3901 2012

[33] M A Petric ldquoSome remarks concerning Ciric-Reich-Rus oper-atorsrdquo Creative Mathematics and Informatics vol 18 no 2 pp188ndash193 2009

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 Abstract and Applied Analysis

such that |119896(119905 119904 119909(119904)) minus 119896(119905 119904 119910(119904))| le 119901(119905 119904)120593(|119909(119904) minus

119910(119904)|) for each 119905 119904 isin [119886 119887] and 119909 le 119910 (ie 119909(119905) le 119910(119905)

for all 119905 isin [119886 119887])

(iii) sup119905isin[119886119887]

int119887

119886119901(119905 119904)119889119904 le 1

(iv) there exists 1199090

isin 119862([119886 119887]R119899) such that 1199090(119905) le

int119887

119886119896(119905 119904 119909

0(119904))119889119904 + 119892(119905) for all 119905 isin [119886 119887]

Then the integral equation (35) has a unique solution in theset 119909 isin 119862([119886 119887]R119899) 119909(119905) le 119909

0(119905) or 119909(119905) ge 119909

0(119905) for all

119905 isin [119886 119887]

Proof Let (119883 sdot infin

) where 119883 = 119862([119886 119887]R119899) and define amapping 119879 119862([119886 119887]R119899) rarr 119862([119886 119887]R119899) by

119879119909 (119905) = int

119887

119886

119896 (119905 119904 119909 (119904)) 119889119904 + 119892 (119905) 119905 isin [119886 119887] (36)

Consider a graph 119866 consisting of 119881(119866) = 119883 and 119864(119866) =

(119909 119910) isin 119883 times 119883 119909(119905) le 119910(119905) for all 119905 isin [119886 119887] From property(i) we observe that the mapping 119879 is nondecreasing thus 119879

preserves edges Furthermore 119866 is a (119862)-graph that is forevery nondecreasing sequence 119909

119899 sub 119883 which converges to

some 119911 isin 119883 then 119909119899(119905) le 119911(119905) for all 119905 isin [119886 119887] Now for every

119909 119910 isin 119883 with (119909 119910) isin 119864(119866) we have

1003816100381610038161003816119879119909 (119905) minus 119879119910 (119905)1003816100381610038161003816

le int

119887

119886

1003816100381610038161003816119896 (119905 119904 119909 (119904)) minus 119896 (119905 119904 119910 (119904))1003816100381610038161003816 119889119904

le int

119887

119886

119901 (119905 119904) 120593 (1003816100381610038161003816119909 (119904) minus 119910 (119904)

1003816100381610038161003816) 119889119904

le 120593 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817infin) int

119887

119886

119901 (119905 119904) 119889119904

(37)

Hence 119889(119879119909 119879119910) le 120593(119889(119909 119910)) From (iv) we have(1199090 1198791199090) isin 119864(119866) so that [119909

0]119866

= 119909 isin 119862([119886 119887]R119899) 119909(119905) le

1199090(119905) or 119909(119905) ge 119909

0(119905) for all 119905 isin [119886 119887] The conclusion follows

fromTheorem 18

Note that Theorem 41 specifies region of solution whichinvokes the novelty of our result

References

[1] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacesrdquo Journal of Functional Analysis vol 30 pp 26ndash37 1989

[2] J Heinonen Lectures on Analysis on Metric Spaces SpringerNew York NY USA 2001

[3] N Bourbaki Topologie Generale Herman Paris France 1974[4] S Czerwik ldquoNonlinear set-valued contraction mappings in

119887-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[5] M Bota A Molnar and C Varga ldquoOn Ekelandrsquos variationalprinciple in 119887-metric spacesrdquo Fixed Point Theory vol 12 no 1pp 21ndash28 2011

[6] M F Bota-Boriceanu andA Petrusel ldquoUlam-Hyers stability foroperatorial equationsrdquo Analele Stiintifice ale Universitatii ldquoAl ICuzardquo din Iasi vol 57 supplement 1 pp 65ndash74 2011

[7] M Pacurar ldquoA fixed point result for 120593-contractions on 119887-metricspaces without the boundedness assumptionrdquo PolytechnicaPosnaniensis InstitutumMathematicum Fasciculi Mathematicino 43 pp 127ndash137 2010

[8] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis Babes-Bolyai vol 41 no 4pp 23ndash27 1996

[9] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[10] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[11] A Petrusel and I A Rus ldquoFixed point theorems in ordered l-spacesrdquo Proceedings of the American Mathematical Society vol134 no 2 pp 411ndash418 2006

[12] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 p 68 1975

[13] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[14] R P Agarwal M A El-Gebeily and D OrsquoRegan ldquoGeneralizedcontractions in partially ordered metric spacesrdquo ApplicableAnalysis vol 87 no 1 pp 109ndash116 2008

[15] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[16] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997

[17] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquoSeminar on Fixed Point Theory vol 3 pp 3ndash9 1993

[18] T P Petru and M Boriceanu ldquoFixed point results for gener-alized 120593-contraction on a set with two metricsrdquo TopologicalMethods in Nonlinear Analysis vol 33 no 2 pp 315ndash326 2009

[19] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 11 pages 2012

[20] A Amini-Harandi ldquoCoupled and tripled fixed point theory inpartially ordered metric spaces with application to initial valueproblemrdquoMathematical and Computer Modelling vol 57 no 9-10 pp 2343ndash2348 2012

[21] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[22] S M A Aleomraninejad Sh Rezapour andN Shahzad ldquoSomefixed point results on ametric space with a graphrdquoTopology andIts Applications vol 159 no 3 pp 659ndash663 2012

[23] M Samreen and T Kamran ldquoFixed point theorems for integralG-contractionsrdquo Fixed Point Theory and Applications vol 2013article 149 2013

[24] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[25] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrix

Abstract and Applied Analysis 9

equationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[26] G Gwozdz-Łukawska and J Jachymski ldquoIFS on a metric spacewith a graph structure and extensions of the Kelisky-Rivlintheoremrdquo Journal of Mathematical Analysis and Applicationsvol 356 no 2 pp 453ndash463 2009

[27] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973

[28] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968

[29] S K Chatterjea ldquoFixed-point theoremsrdquo Doklady BolgarskoıAkademii Nauk vol 25 pp 727ndash730 1972

[30] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003

[31] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010

[32] F Bojor ldquoFixed point theorems for Reich type contractionson metric spaces with a graphrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 9 pp 3895ndash3901 2012

[33] M A Petric ldquoSome remarks concerning Ciric-Reich-Rus oper-atorsrdquo Creative Mathematics and Informatics vol 18 no 2 pp188ndash193 2009

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

Abstract and Applied Analysis 9

equationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[26] G Gwozdz-Łukawska and J Jachymski ldquoIFS on a metric spacewith a graph structure and extensions of the Kelisky-Rivlintheoremrdquo Journal of Mathematical Analysis and Applicationsvol 356 no 2 pp 453ndash463 2009

[27] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973

[28] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968

[29] S K Chatterjea ldquoFixed-point theoremsrdquo Doklady BolgarskoıAkademii Nauk vol 25 pp 727ndash730 1972

[30] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003

[31] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010

[32] F Bojor ldquoFixed point theorems for Reich type contractionson metric spaces with a graphrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 9 pp 3895ndash3901 2012

[33] M A Petric ldquoSome remarks concerning Ciric-Reich-Rus oper-atorsrdquo Creative Mathematics and Informatics vol 18 no 2 pp188ndash193 2009

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