Research Article Generalized Contraction and Invariant Approximation …downloads.hindawi.com/journals/aaa/2014/391952.pdf · 2019. 7. 31. · Research Article Generalized Contraction

Research ArticleGeneralized Contraction and Invariant Approximation Resultson Nonconvex Subsets of Normed Spaces

Mujahid Abbas1 Basit Ali2 and Salvador Romaguera3

1 Department of Mathematics and Applied Mathematics University of Pretoria Lynnwood Road Pretoria 0002 South Africa2Department of Mathematics Syed Babar Ali School of Science and Engineering Lahore University of Management SciencesLahore 54792 Pakistan

3 Instituto Universitario de Matematica Pura y Aplicada Universitat Politecnica de Valencia Camı de Vera sn 46022 Valencia Spain

Correspondence should be addressed to Salvador Romaguera sromaguematupves

Received 6 December 2013 Accepted 18 January 2014 Published 26 February 2014

Academic Editor Wei-Shih Du

Copyright copy 2014 Mujahid Abbas 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

Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces asa generalization of Banach contraction principle In this paper we introduce a notion of generalized F-contraction mappingswhich is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spacesSome theorems on invariant approximations in normed linear spaces are also deduced Our results extend unify and generalizecomparable results in the literature

1 Introduction and Preliminaries

One of the most basic and important results in metric fixedpoint theory is the Banach contraction principle due toBanach [1] It states that if (119883 119889) is a complete metric spaceand 119891 119883 rarr 119883 satisfies

119889 (119891119909 119891119910) le 119896119889 (119909 119910) (1)

for all 119909 119910 isin 119883 with 119896 isin (0 1) then 119891 has a unique fixedpoint This theorem that has been extended in many direc-tions (see eg [2ndash6]) has many applications in mathematicsand other related disciplines as well (see eg [7ndash9])

Meinardus [10] and Brosowski [11] employed fixed pointtheory to obtain invariant approximation results in normedlinear spaces A number of authors generalized their results(see [12ndash18] and the references therein) On the other handDotson [19] extended Banachrsquos contraction principle fornonexpansive mappings on star-shaped subsets of Banachspaces and proved Brosowski-Meinardus type theorems oninvariant approximations L A Khan and A R Khan [20]generalized Dotsonrsquos results on star shaped subsets of p-normed spaces

Recently Wardowski [21] introduced a new contractivemapping called an 119865-contraction and proved some fixedpoint theorems in complete metric spaces In this paper weintroduce a notion of generalized F-contraction mappingswhich is used to prove a fixed point result for generalizednonexpansive mappings on star shaped subsets of normedlinear spaces Some theorems on invariant approximations innormed linear spaces are deduced Our results extend unifyand generalize comparable results in [10 11 19 20] Someillustrative examples are also presented

Next we give some definitions which will be used in thesequel The letters R

+ R will denote the set of positive real

numbers and the set of real numbers respectivelyLet ϝ be the collection of all mappings 119865 R

+rarr R

which satisfy the following conditions

(C1) 119865 is strictly increasing that is for all 120572 120573 isin R+such

that 120572 lt 120573 rArr 119865(120572) lt 119865(120573)

(C2) for every sequence 120572119899119899ge1

of positive numberslim119899rarrinfin

120572119899= 0 if and only if lim

119899rarrinfin119865(120572119899) = minusinfin

(C3) there exists a 119896 isin (0 1) such that lim120572rarr0

+120572119896119865(120572) = 0

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 391952 5 pageshttpdxdoiorg1011552014391952

2 Abstract and Applied Analysis

Definition 1 (see [21]) Let (119883 119889) be ametric space and 119865 isin ϝA mapping 119891 119883 rarr 119883 is said to be an 119865-contraction on 119883

if there exists a 120591 gt 0 such that

119889 (119891119909 119891119910) gt 0 997904rArr 120591 + 119865 (119889 (119891119909 119891119910)) le 119865 (119889 (119909 119910)) (2)

for all 119909 119910 isin 119883

Remark 2 (see [21]) Every 119865-contraction mapping is contin-uous

Motivated by thework ofWardowski [21] and byTheorem4 of [22] we give the following definition

Definition 3 Let (119883 119889) be a metric space and 119865 isin ϝ Amapping119891 119883 rarr 119883 is said to be a generalized119865-contractionif there exists a 120591 gt 0 such that

119889 (119891119909 1198912

119909) gt 0 997904rArr 120591 + 119865 (119889 (119891119909 1198912

119909)) le 119865 (119889 (119909 119891119909))

(3)

for all 119909 isin 119883

Definition 4 Let (119883 119889) be a metric space and 119865 isin ϝ Amapping 119891 119883 rarr 119883 is said to be 119865-nonexpansive if

119889 (119891119909 119891119910) gt 0 997904rArr 119865 (119889 (119891119909 119891119910)) le 119865 (119889 (119909 119910)) (4)

for all 119909 119910 isin 119883

Remark 5 It follows from condition (C1) that if 119865 isin ϝ

and 119891 is an 119865-nonexpansive self-mapping of a metric space(119883 119889) then 119891 is nonexpansive (recall that 119891 is nonexpansiveprovided that 119889 (119891119909 119891119910) le 119889 (119909 119910) for all 119909 119910 isin 119883)Conversely it is clear by (C1) that if 119891 is a nonexpansive self-mapping of a metric space (119883 119889) then 119891 is 119865-nonexpansivefor all 119865 isin ϝ

By considering different choices of mappings 119865 in (2) (3)and (4) we obtain a variety of contractions For details werefer to [21] and the following examples

Example 6 Let (119883 119889) be a metric space 119865 isin ϝ and let 119866

R+

rarr R be given by 119866 (120572) = 119865 (120572) minus 120591 where 120591 gt 0 Itis clear that 119866 isin ϝ Now if 119891 119883 rarr 119883 is a generalized119865-contraction then it is a generalized 119866-contraction becausefor any 119909 119910 isin 119883 with 119889 (119891119909 1198912119909) gt 0 we have

120591 + 119866 (119889 (119891119909 1198912

119909)) = 119865 (119889 (119891119909 1198912

119909))

le 119865 (119889 (119909 119891119909)) minus 120591 = 119866 (119889 (119909 119891119909))

(5)

Similarly if 119891 is an 119865-contraction then it is a 119866-contractionFurthermore if 119891 is 119865-nonexpansive then

119866 (119889 (119891119909 119891119910)) = 119865 (119889 (119891119909 119891119910)) minus 120591

le 119865 (119889 (119909 119910)) minus 2120591 le 119866 (119889 (119909 119910)) minus 120591

le 119866 (119889 (119909 119910))

(6)

whenever 119889 (119891119909 119891119910) gt 0 which shows that 119891 is 119866-nonexpansive Finally note that taking 119866 (120572) = ln (120572) in (6)we deduce that 119891 is nonexpansive

Example 7 Let (119883 119889) be a metric space let 1198651 R+rarr R be

given by 1198651(120572) = ln (120572) and let 119891 119883 rarr 119883 be a generalized

119865-contraction Since 1198651isin ϝ then (3) becomes

120591 + ln (119889 (119891119909 1198912119909)) le ln (119889 (119909 119891119909)) (7)

whenever 119889 (119891119909 1198912119909) gt 0 which implies

ln119889 (119891119909 1198912119909)

119889 (119909 119891119909)le minus120591 that is

119889 (119891119909 1198912119909)

119889 (119909 119891119909)le 119890minus120591

(8)

and thus 119889 (119891119909 1198912119909) le 119890minus120591119889 (119909 119891119909) Hence our definition ismore general than those given in [18 20]

If we take 1198652(120572) = ln(120572) + 120572 it is clear that 119865

2isin ϝ and

then (2) becomes

120591 + ln (119889 (119891119909 1198912119909)) + 119889 (119891119909 1198912

119909) le ln (119889 (119909 119891119909))

+ 119889 (119909 119891119909)

(9)

whenever 119889(119891119909 1198912119909) gt 0 which implies that

119889 (119891119909 1198912119909)

119889 (119909 119891119909)119890119889(119891119909119891

2119909)minus119889(119909119891119909)

le 119890minus120591

(10)

that is

119889 (119891119909 1198912

119909) le119890minus120591

119890119889(1198911199091198912119909)minus119889(119909119891119909)

119889 (119909 119891119909) (11)

Definition 8 Let 119862 be a closed subset of metric space (119883 119889)Then 119891 119862 rarr 119862 is called compact if for every boundedsubset 119860 of 119862 119891(119860) is compact in 119862

Definition 9 If 119891 119883 rarr 119883 is a mapping with 119891(119862) sube 119862then 119862 is called an 119891-invariant subset of119883

Definition 10 Let 119862 be a subset of metric space (119883 119889) Asusual for any 119909 isin 119883 we define

119889 (119909 119862) = inf 119889 (119909 119910) 119910 isin 119862

119875119862(119909) = 119910 isin 119862 119889 (119909 119910) = 119889 (119909 119862)

(12)

119875119862(119909) is called the set of best approximations of 119909 from 119862 If

for each 119909 isin 119883 119875119862(119909) is nonempty then119862 is called proximal

Observe that if 119862 is closed then 119875119862(119909) is also closed

Definition 11 Let 119864 be a linear space over R A subset 119862 of 119864is called star-shaped if there exists at least one point 119911 isin 119862

such that 119905119911 + (1 minus 119905)119909 isin 119862 for all 119909 isin 119862 and 0 lt 119905 lt 1 In thiscase 119911 is called a star centre of 119862

Let (119883 119889) be a metric space 119862 a closed subset of 119883 and119891 119862 rarr 119862 a self-mapping For each 119909 isin 119862 the set

Abstract and Applied Analysis 3

119874(119909) = 119909 119891119909 119891119899119909 is called the orbit of 119909 (compare[23]) The mapping 119891 is called orbitally continuous at 119901 iflim119899rarrinfin

119891119899119909 = 119901 implies lim119899rarrinfin

119891119899+1119909 = 119891119901 and 119891 isorbitally continuous on a set 119862 if 119891 is orbitally continuousfor all 119901 isin 119862

2 Main Results

In the following a normed linear space (119864 sdot ) will besimply denoted by 119864 if no confusion arises Furthermoreby a complete subset of a normed linear space 119864 we willmean a subset 119860 of 119864 such that the restriction to 119860 of themetric induced on119864 by its norm is complete Of course everycomplete subset of a normed linear space is closed and everyclosed subset of a Banach space is complete

Our main result (Theorem 13 below) will be proved withthe help of the following re-formulation ofTheorem 4 of [22]

Theorem 12 (see [22]) Let (119883 119889) be a complete metric space119865 isin ϝ and 119891 119883 rarr 119883 an orbitally continuous generalized119865-contraction Then 119891 has a fixed point

Theorem 13 Let 119864 be a normed linear space 119862 a completeand star-shaped subset of 119864 and 119865 isin ϝ If 119891 119862 rarr 119862 is an119865-nonexpansive mapping and 119891(119862) is compact then 119891 has afixed point

Proof We first note that by Remark 5 119891 is nonexpansive on119862 so it is continuous on 119862

Now let 119911 be a star centre of 119862 For each 119899 ge 1 define119891119899 119862 rarr 119862 by

119891119899119909 = (1 minus 119896

119899) 119911 + 119896

119899119891119909 (13)

for all 119909 isin 119862 where 0 lt 119896119899lt 1 and lim

119899rarrinfin119896119899= 1 From

the fact that 119891 is continuous on 119862 it immediately follows thateach 119891

119899is continuous on 119862

For any fixed 119899 ge 1 and any 119909 isin 119862 we have

119865 (10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) = 119865 (

1003817100381710038171003817(1 minus 119896119899) 119911 + 119896

119899119891119909

minus119891119899((1 minus 119896

119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

= 119865 (1003817100381710038171003817(1 minus 119896

119899) 119911 + 119896

119899119891119909 minus (1 minus 119896

119899) 119911

minus119896119899119891 ((1 minus 119896

119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

= 119865 (1003817100381710038171003817119896119899 (119891119909 minus 119891 ((1 minus 119896

119899) 119911 + 119896

119899119891119909))

1003817100381710038171003817)

(14)

Since 119865 is strictly increasing with 119896119899lt 1 for each 119899 ge 1

and 119891 is 119865-nonexpansive we have

119865 (10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) lt 119865 (

1003817100381710038171003817119891119909 minus 119891 ((1 minus 119896119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

lt 119865 (1003817100381710038171003817119909 minus ((1 minus 119896

119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

= 119865 (1003817100381710038171003817119909 minus 119891

1198991199091003817100381710038171003817)

(15)

Hence

119865 (1003817100381710038171003817119909 minus 119891

1198991199091003817100381710038171003817) minus 119865 (

10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) gt 0 (16)

This implies that there exists 120591119899gt 0 such that

0 lt 120591119899le 119865 (

1003817100381710038171003817119909 minus 1198911198991199091003817100381710038171003817) minus 119865 (

10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) (17)

Therefore

120591119899+ 119865 (

10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) le 119865 (

1003817100381710038171003817119909 minus 1198911198991199091003817100381710038171003817) (18)

Hence 119891119899is a generalized 119865-contraction for each 119899 ge 1 By

Theorem 12 for each 119899 ge 1 there exists 119909119899isin 119862 such that

119891119899119909119899= 119909119899 Since 119891(119862) is compact there exist a subsequence

119891119909119899119894

119894ge1

of the sequence 119891119909119899119899ge1

and an 119909 isin 119891(119862) such that

119909 = lim119894rarrinfin

119891119909119899119894

(19)

In fact 119909 isin 119862 because 119891(119862) sube 119862 and 119862 is closedSince lim

119894rarrinfin119896119899119894

= 1 and 119909119899119894

= 119891119899119894

119909119899119894

for all 119894 ge 1 wededuce that


119891119909119899119894

= lim119894rarrinfin

((1 minus 119896119899119894

) 119911 + 119896119899119894

119891119909119899119894

)


119891119899119894

119909119899119894


119909119899119894

(20)

Therefore

119891119909 = lim119894rarrinfin

119891119909119899119894

(21)

We conclude that 119909 = 119891119909

Remark 14 Note that by Remark 5 we can restate thepreceding theorem as follows Let 119864 be a normed linear space(Banach space resp) and 119862 a complete (closed resp) andstar-shaped subset of 119864 If 119891 119862 rarr 119862 is a nonexpansivemapping and 119891(119862) is compact then 119891 has a fixed point

The following is an example where we can apply Theo-rem 13 to every 119865 isin ϝ

Example 15 Let ℓ1be the linear space of all summable

sequences of real numbers Then the pair (ℓ1 sdot 1) is a

(classical) Banach space where sdot 1is the norm on ℓ

1such

that for each x = 119909119899119899ge1

isin ℓ1

x1=

infin

sum119899=1

10038161003816100381610038161199091198991003816100381610038161003816 (22)

In the following any element x = 119909119899119899ge1

of ℓ1will be also

denoted as (1199091 1199092 1199093 119909

119899 )

Let 119862 be the closed unit ball of (ℓ1 sdot 1) that is

119862 = x isin ℓ1 x1le 1 (23)

It is well known that 119862 is a (noncompact) closed subset of(ℓ1 sdot 1) Moreover 119862 is star-shaped with z = 0 a star center

of 119862Now let 119896 isin (0 1] be constant and define 119891 119862 rarr 119862 as

119891x = (1199091 1198961199092 0 0 ) (24)


for all x = (119909119899)119899ge1

isin ℓ1 Clearly 119891 is nonexpansive on 119862 and

hence it is 119865-nonexpansive for any 119865 isin ϝ by Remark 5Furthermore 119891(119862) is homeomorphic to the compact

subset of R2

(119906 V) isin R2

|119906| +1003816100381610038161003816100381610038161003816

V119896

1003816100381610038161003816100381610038161003816le 1 (25)

so that 119891(119862) is compact Hence 119891(119862) = 119891(119862) and thus 119891(119862)is compact We have shown that all conditions ofTheorem 13(compare Remark 14) are satisfied Thus 119891 has a fixed pointIn fact the fixed points of 119891 are the elements x = 119909

119899119899ge1

of119862 such that 119909

119899= 0 whenever 119899 ge 2

Now we give an example of a discontinuous self-mapping119891 on a compact metric space which is a generalized 119865-contraction but not an 119865-contraction So the class of gen-eralized 119865-contraction mappings is bigger than the class of119865-contraction

Example 16 Let 119883 = R and let 119862 = [0 1] be endowed withusual metric Define a mapping 119891 119862 rarr 119862 as 119891(119909) = 1 if119909 isin 0 1 and 119891(119909) = 1199092 if 119909 isin (0 1)

Let 119865 R+

rarr R be defined by 119865119909 = ln119909 for 119909 isin (0 1)Note that

ln 2 + 119865 (119889 (119891119909 1198912

119909)) le 119865 (119889 (119909 119891119909)) (26)

is satisfied for all 119909 isin 119862 whenever 119889 (119891119909 1198912119909) gt 0 Hence119891 is a generalized 119865-contraction Clearly 119891 is not continuousat 119909 = 1 and at 119909 = 0 Hence 119891 is not an 119865-contraction (seeRemark 21 in [21] ) or let 119909 = 0 and 119910 = 12 then

119865(119889(1198910 1198911

2)) = 119865(119889(1

1

4))

= 119865(3

4) = ln(3

4) = ln 3 minus 2 ln 2

(27)

while

119865(119889(01

2)) = 119865(

1

2) = ln(1

2) = minus ln 2 (28)

and thus

119865(119889(01

2)) minus 119865(119889(1198910 119891

1

2)) = ln 2 minus ln 3 lt 0 (29)

Hence there does not exist any 120591 gt 0 such that for 119909 = 0 and119910 = 12

120591 + 119865 (119889 (119891119909 119891119910)) le 119865 (119889 (119909 119910)) (30)

is satisfied Now let 1199090= 0 then 119891(0) = 1 and 119891

2(0) = 1

Hence the orbit of 1199090is the set 119874 (0) = 0 1 1 = 119874 (0)

which is compact and

lim119899rarrinfin

119891119899

(0) = 1 lim119899rarrinfin

119891119899+1

(0) = 1198911 = 1 (31)

that is 119891 is orbitally continuous at 1 Hence byTheorem 12 119891has a fixed point (in fact 119909 = 1 is the only fixed point of 119891)

In the last part of the paper we discuss nonemptiness andexistence of fixed points for the set of best approximationsof closed subsets of metric spaces and of normed spacesrespectively

Theorem 17 Let (119883 119889) be a metric space Let 119865 isin ϝ be acontinuous mapping and let 119891 119883 rarr 119883 be 119865-nonexpansivewith a fixed point 119906 isin 119883 If 119862 is a closed 119891-invariant subsetof 119883 such that 119891 is compact on 119862 then the set 119875

119862(119906) of best

approximations is nonempty

Proof Let 119903 = 119889 (119906 119862) Then there is a sequence 119910119899119899ge1

in 119862 such that lim119899rarrinfin

119889 (119906 119910119899) = 119903 Since 119910

119899 119899 ge 1

is a bounded subset of 119862 and 119891 is compact on 119862 the set119891119910119899 119899 ge 1 is a compact subset of 119862 and so there exist a

subsequence 119891119910119899119894

119894ge1

of 119891119910119899119899ge1

and an 119909 isin 119862 such thatlim119894rarrinfin

119891119910119899119894

= 119909 Now

119865 (119903) le 119865 (119889 (119906 119909)) = 119865 ( lim119894rarrinfin

119889 (119891119906 119891119910119899119894

))


119865 (119889 (119891119906 119891119910119899119894

))

le lim119894rarrinfin

119865 (119889 (119906 119910119899119894

))

= 119865( lim119894rarrinfin

119889 (119906 119910119899119894

)) = 119865 (119903)

(32)

This implies

119865 (119903) = 119865 (119889 (119906 119909)) (33)

Since119865 is strictly increasing we get 119903 = 119889 (119906 119909) Hence119875119862(119906)

is nonempty

As an application of Theorems 13 and 17 we deduce thefollowing

Theorem 18 Let 119864 be a normed linear space Let 119865 isin ϝ be acontinuous mapping and let 119891 119864 rarr 119864 be 119865-nonexpansivewith a fixed point 119906 isin 119864 If 119862 is a complete 119891-invariant subsetof 119864 such that 119891 is compact on 119862 and 119875

119862(119906) is a star-shaped

set then 119891 has a fixed point in 119875119862(119906)

Proof By Theorem 17 119875119862(119906) is nonempty We show that

119875119862(119906) is 119891-invariant To this end let 119910 isin 119875

119862(119906) and set

119903 = 119889 (119906 119862) then

119865 (119903) le 119865 (119889 (119906 119891119910)) = 119865 (119889 (119891119906 119891119910))

le 119865 (119889 (119906 119910)) = 119865 (119903) (34)

This implies

119865 (119903) = 119865 (119889 (119906 119891119910)) (35)

Since 119865 is strictly increasing we get 119903 = 119889 (119906 119891119910) So 119891119910 isin

119875119862(119906) This proves that 119875

119862(119906) is 119891-invariant so 119891 119875

119862(119906) rarr

119875119862(119906) is 119865-nonexpansive Now observe that if 119862 is complete

then 119875119862(119906) is also complete Hence 119875

119862(119906) is star-shaped and

complete and 119891(119875119862(119906)) is compact so byTheorem 13 there

exists 119909 isin 119875119862(119906) such that 119891119909 = 119909


Remark 19 As in the case of Theorem 13 (see Remark 14)the preceding theorem can be restated as follows Let 119864 be anormed linear space (Banach space resp) and let 119891 119864 rarr 119864

be nonexpansive with a fixed point 119906 isin 119864 If 119862 is a complete(closed resp) 119891-invariant subset of 119864 such that 119891 is compacton 119862 and 119875

119862(119906) is a star-shaped set then 119891 has a fixed point

in 119875119862(119906)

We conclude the paper illustrating Theorem 18 with thefollowing example

Example 20 Let (ℓ1 sdot 1) be the Banach space of Exam-

ple 15 Define 119891 ℓ1rarr ℓ1as

119891x = (1199091 1198961199092 0 0 ) (36)

for all x = 119909119899119899ge1

isin ℓ1 with 119896 isin (0 1] Let 119865 isin

ϝ be continuous Since 119891 is nonexpansive it follows fromRemark 5 that it is 119865-nonexpansive Of course 119891 has fixedpoints In fact

Fix (119891) = x = 119909119899119899ge1

isin ℓ1 119909119899= 0 forall119899 ge 2 (37)

As in Example 15 let119862 be the closed unit ball of (ℓ1 sdot 1)

We know that 119862 is a closed and thus complete 119891-invariantsubset of 119864 such that 119891 is compact on 119862

Then if we choose x = 119909119899119899ge1

isin Fix(119891) such that|1199091| gt 1 we deduce that 119875

119862(x) = (1 0 0 0 ) if 119909

1gt 1

and 119875119862(x) = (minus1 0 0 0 ) if 119909

1lt minus1 Therefore 119875

119862(119906)

is trivially star-shaped Thus all conditions of Theorem 18(compare Remark 19) have been verified

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to the referees for their valuablecomments and suggestions and in particular to one of themfor calling our attention on the crucial fact stated in thefirst part of Remark 5 and for the elegant reformulationof Theorem 13 stated in Remark 14 Salvador Romagueraacknowledges the support of the Universitat Politecnica deValencia Grant PAID-06-12-SP20120471

References

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

[2] I Arandjelovic Z Kadelburg and S Radenovic ldquoBoyd-Wong-type common fixed point results in conemetric spacesrdquoAppliedMathematics and Computation vol 217 no 17 pp 7167ndash71712011

[3] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 pp458ndash464 1969

[4] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[5] E Rakotch ldquoA note on contractivemappingsrdquo Proceedings of theAmerican Mathematical Society vol 13 pp 459ndash465 1962

[6] E Tarafdar ldquoAn approach to fixed-point theorems on uniformspacesrdquo Transactions of the AmericanMathematical Society vol191 pp 209ndash225 1974

[7] J G Dix and G L Karakostas ldquoA fixed-point theorem forS-type operators on Banach spaces and its applications toboundary-value problemsrdquoNonlinearAnalysisTheoryMethodsamp Applications vol 71 no 9 pp 3872ndash3880 2009

[8] K Latrach M A Taoudi and A Zeghal ldquoSome fixed pointtheorems of the Schauder and the Krasnoselrsquoskii type and appli-cation to nonlinear transport equationsrdquo Journal of DifferentialEquations vol 221 no 1 pp 256ndash271 2006

[9] C Rousseau ldquoPoint fixe de Banachrdquo Accromath 5 hiver-printemps 2010 (in French) httpwwwaccromathca

[10] G Meinardus ldquoInvarianz bei linearen ApproximationenrdquoArchive for RationalMechanics andAnalysis vol 14 pp 301ndash3031963

[11] B Brosowski ldquoFixpunktsatze in der ApproximationstheorierdquoMathematica vol 11 no 34 pp 195ndash220 1969

[12] L Habiniak ldquoFixed point theorems and invariant approxima-tionsrdquo Journal of Approximation Theory vol 56 no 3 pp 241ndash244 1989

[13] T L Hicks and M D Humphries ldquoA note on fixed-pointtheoremsrdquo Journal of Approximation Theory vol 34 no 3 pp221ndash225 1982

[14] N Hussain D OrsquoRegan and R P Agarwal ldquoCommon fixedpoint and invariant approximation results on non-star-shapeddomainrdquoGeorgianMathematical Journal vol 12 no 4 pp 659ndash669 2005

[15] T D Narang ldquoApplications of fixed point theorems to approx-imation theoryrdquo Matematicki Vesnik vol 36 no 1 pp 69ndash751984

[16] S P Singh ldquoAn application of a fixed-point theorem to approx-imation theoryrdquo Journal of ApproximationTheory vol 25 no 1pp 89ndash90 1979

[17] A Smoluk ldquoInvariant approximationsrdquo Matematyka Stosow-ana vol 17 pp 17ndash22 1981

[18] P V Subrahmanyam ldquoAn application of a fixed point theoremto best approximationrdquo Journal of Approximation Theory vol20 no 2 pp 165ndash172 1977

[19] W G Dotson ldquoFixed point theorems for non-expansive map-pings on star-shaped subsets of Banach spacesrdquo Journal of theLondon Mathematical Society vol 4 pp 408ndash410 1972

[20] L A Khan and A R Khan ldquoAn extension of Brosowski-Meinardus theorem on invariant approximationrdquo Approxima-tion Theory and its Applications vol 11 no 4 pp 1ndash5 1995

[21] D Wardowski ldquoFixed points of a new type of contractivemappings in complete metric spacesrdquo Fixed Point Theory andApplications vol 2012 article 94 6 pages 2012

[22] M Abbas B Ali and S Romaguera ldquoFixed and periodic pointsof generalized contractions inmetric spacesrdquoFixed PointTheoryand Applications vol 2013 article 243 11 pages 2013

[23] T L Hicks and B E Rhoades ldquoA Banach type fixed-pointtheoremrdquo Mathematica Japonica vol 24 no 3 pp 327ndash3301979

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Definition 1 (see [21]) Let (119883 119889) be ametric space and 119865 isin ϝA mapping 119891 119883 rarr 119883 is said to be an 119865-contraction on 119883

if there exists a 120591 gt 0 such that

119889 (119891119909 119891119910) gt 0 997904rArr 120591 + 119865 (119889 (119891119909 119891119910)) le 119865 (119889 (119909 119910)) (2)

for all 119909 119910 isin 119883

Remark 2 (see [21]) Every 119865-contraction mapping is contin-uous

Motivated by thework ofWardowski [21] and byTheorem4 of [22] we give the following definition

Definition 3 Let (119883 119889) be a metric space and 119865 isin ϝ Amapping119891 119883 rarr 119883 is said to be a generalized119865-contractionif there exists a 120591 gt 0 such that

119889 (119891119909 1198912

119909) gt 0 997904rArr 120591 + 119865 (119889 (119891119909 1198912

119909)) le 119865 (119889 (119909 119891119909))

(3)

for all 119909 isin 119883

Definition 4 Let (119883 119889) be a metric space and 119865 isin ϝ Amapping 119891 119883 rarr 119883 is said to be 119865-nonexpansive if

119889 (119891119909 119891119910) gt 0 997904rArr 119865 (119889 (119891119909 119891119910)) le 119865 (119889 (119909 119910)) (4)

for all 119909 119910 isin 119883

Remark 5 It follows from condition (C1) that if 119865 isin ϝ

and 119891 is an 119865-nonexpansive self-mapping of a metric space(119883 119889) then 119891 is nonexpansive (recall that 119891 is nonexpansiveprovided that 119889 (119891119909 119891119910) le 119889 (119909 119910) for all 119909 119910 isin 119883)Conversely it is clear by (C1) that if 119891 is a nonexpansive self-mapping of a metric space (119883 119889) then 119891 is 119865-nonexpansivefor all 119865 isin ϝ

By considering different choices of mappings 119865 in (2) (3)and (4) we obtain a variety of contractions For details werefer to [21] and the following examples

Example 6 Let (119883 119889) be a metric space 119865 isin ϝ and let 119866

R+

rarr R be given by 119866 (120572) = 119865 (120572) minus 120591 where 120591 gt 0 Itis clear that 119866 isin ϝ Now if 119891 119883 rarr 119883 is a generalized119865-contraction then it is a generalized 119866-contraction becausefor any 119909 119910 isin 119883 with 119889 (119891119909 1198912119909) gt 0 we have

120591 + 119866 (119889 (119891119909 1198912

119909)) = 119865 (119889 (119891119909 1198912

119909))

le 119865 (119889 (119909 119891119909)) minus 120591 = 119866 (119889 (119909 119891119909))

(5)

Similarly if 119891 is an 119865-contraction then it is a 119866-contractionFurthermore if 119891 is 119865-nonexpansive then

119866 (119889 (119891119909 119891119910)) = 119865 (119889 (119891119909 119891119910)) minus 120591

le 119865 (119889 (119909 119910)) minus 2120591 le 119866 (119889 (119909 119910)) minus 120591

le 119866 (119889 (119909 119910))

(6)

whenever 119889 (119891119909 119891119910) gt 0 which shows that 119891 is 119866-nonexpansive Finally note that taking 119866 (120572) = ln (120572) in (6)we deduce that 119891 is nonexpansive

Example 7 Let (119883 119889) be a metric space let 1198651 R+rarr R be

given by 1198651(120572) = ln (120572) and let 119891 119883 rarr 119883 be a generalized

119865-contraction Since 1198651isin ϝ then (3) becomes

120591 + ln (119889 (119891119909 1198912119909)) le ln (119889 (119909 119891119909)) (7)

whenever 119889 (119891119909 1198912119909) gt 0 which implies

ln119889 (119891119909 1198912119909)

119889 (119909 119891119909)le minus120591 that is

119889 (119891119909 1198912119909)

119889 (119909 119891119909)le 119890minus120591

(8)

and thus 119889 (119891119909 1198912119909) le 119890minus120591119889 (119909 119891119909) Hence our definition ismore general than those given in [18 20]

If we take 1198652(120572) = ln(120572) + 120572 it is clear that 119865

2isin ϝ and

then (2) becomes

120591 + ln (119889 (119891119909 1198912119909)) + 119889 (119891119909 1198912

119909) le ln (119889 (119909 119891119909))

+ 119889 (119909 119891119909)

(9)

whenever 119889(119891119909 1198912119909) gt 0 which implies that

119889 (119891119909 1198912119909)

119889 (119909 119891119909)119890119889(119891119909119891

2119909)minus119889(119909119891119909)

le 119890minus120591

(10)

that is

119889 (119891119909 1198912

119909) le119890minus120591

119890119889(1198911199091198912119909)minus119889(119909119891119909)

119889 (119909 119891119909) (11)

Definition 8 Let 119862 be a closed subset of metric space (119883 119889)Then 119891 119862 rarr 119862 is called compact if for every boundedsubset 119860 of 119862 119891(119860) is compact in 119862

Definition 9 If 119891 119883 rarr 119883 is a mapping with 119891(119862) sube 119862then 119862 is called an 119891-invariant subset of119883

Definition 10 Let 119862 be a subset of metric space (119883 119889) Asusual for any 119909 isin 119883 we define

119889 (119909 119862) = inf 119889 (119909 119910) 119910 isin 119862

119875119862(119909) = 119910 isin 119862 119889 (119909 119910) = 119889 (119909 119862)

(12)

119875119862(119909) is called the set of best approximations of 119909 from 119862 If

for each 119909 isin 119883 119875119862(119909) is nonempty then119862 is called proximal

Observe that if 119862 is closed then 119875119862(119909) is also closed

Definition 11 Let 119864 be a linear space over R A subset 119862 of 119864is called star-shaped if there exists at least one point 119911 isin 119862

such that 119905119911 + (1 minus 119905)119909 isin 119862 for all 119909 isin 119862 and 0 lt 119905 lt 1 In thiscase 119911 is called a star centre of 119862

Let (119883 119889) be a metric space 119862 a closed subset of 119883 and119891 119862 rarr 119862 a self-mapping For each 119909 isin 119862 the set





2 Main Results







119891119899119909 = (1 minus 119896

119899) 119911 + 119896

119899119891119909 (13)


119899rarrinfin119896119899= 1 From




119865 (10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) = 119865 (

1003817100381710038171003817(1 minus 119896119899) 119911 + 119896

119899119891119909

minus119891119899((1 minus 119896

119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

= 119865 (1003817100381710038171003817(1 minus 119896

119899) 119911 + 119896

119899119891119909 minus (1 minus 119896

119899) 119911

minus119896119899119891 ((1 minus 119896

119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

= 119865 (1003817100381710038171003817119896119899 (119891119909 minus 119891 ((1 minus 119896

119899) 119911 + 119896

119899119891119909))

1003817100381710038171003817)

(14)



119865 (10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) lt 119865 (

1003817100381710038171003817119891119909 minus 119891 ((1 minus 119896119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

lt 119865 (1003817100381710038171003817119909 minus ((1 minus 119896

119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

= 119865 (1003817100381710038171003817119909 minus 119891

1198991199091003817100381710038171003817)

(15)

Hence

119865 (1003817100381710038171003817119909 minus 119891

1198991199091003817100381710038171003817) minus 119865 (

10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) gt 0 (16)


0 lt 120591119899le 119865 (

1003817100381710038171003817119909 minus 1198911198991199091003817100381710038171003817) minus 119865 (

10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) (17)

Therefore

120591119899+ 119865 (

10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) le 119865 (

1003817100381710038171003817119909 minus 1198911198991199091003817100381710038171003817) (18)




119891119909119899119894

119894ge1




119891119909119899119894

(19)


119894rarrinfin119896119899119894

= 1 and 119909119899119894

= 119891119899119894

119909119899119894



119891119909119899119894


((1 minus 119896119899119894

) 119911 + 119896119899119894

119891119909119899119894

)


119891119899119894

119909119899119894


119909119899119894

(20)

Therefore

119891119909 = lim119894rarrinfin

119891119909119899119894

(21)

We conclude that 119909 = 119891119909






1such

that for each x = 119909119899119899ge1

isin ℓ1

x1=

infin

sum119899=1

10038161003816100381610038161199091198991003816100381610038161003816 (22)


of ℓ1will be also

denoted as (1199091 1199092 1199093 119909

119899 )


119862 = x isin ℓ1 x1le 1 (23)



119891x = (1199091 1198961199092 0 0 ) (24)


for all x = (119909119899)119899ge1



subset of R2

(119906 V) isin R2

|119906| +1003816100381610038161003816100381610038161003816

V119896

1003816100381610038161003816100381610038161003816le 1 (25)


119899119899ge1


119899= 0 whenever 119899 ge 2



Let 119865 R+


ln 2 + 119865 (119889 (119891119909 1198912

119909)) le 119865 (119889 (119909 119891119909)) (26)


119865(119889(1198910 1198911

2)) = 119865(119889(1

1

4))

= 119865(3

4) = ln(3


(27)

while

119865(119889(01

2)) = 119865(

1

2) = ln(1

2) = minus ln 2 (28)

and thus

119865(119889(01

2)) minus 119865(119889(1198910 119891

1

2)) = ln 2 minus ln 3 lt 0 (29)


120591 + 119865 (119889 (119891119909 119891119910)) le 119865 (119889 (119909 119910)) (30)


2(0) = 1



lim119899rarrinfin

119891119899


119891119899+1

(0) = 1198911 = 1 (31)




119862(119906) of best




119889 (119906 119910119899) = 119903 Since 119910

119899 119899 ge 1


subsequence 119891119910119899119894

119894ge1

of 119891119910119899119899ge1


119891119910119899119894

= 119909 Now

119865 (119903) le 119865 (119889 (119906 119909)) = 119865 ( lim119894rarrinfin

119889 (119891119906 119891119910119899119894

))


119865 (119889 (119891119906 119891119910119899119894

))


119865 (119889 (119906 119910119899119894

))


119889 (119906 119910119899119894

)) = 119865 (119903)

(32)

This implies

119865 (119903) = 119865 (119889 (119906 119909)) (33)


is nonempty







119862(119906) and set

119903 = 119889 (119906 119862) then

119865 (119903) le 119865 (119889 (119906 119891119910)) = 119865 (119889 (119891119906 119891119910))

le 119865 (119889 (119906 119910)) = 119865 (119903) (34)

This implies

119865 (119903) = 119865 (119889 (119906 119891119910)) (35)


119875119862(119906) This proves that 119875

119862(119906) is 119891-invariant so 119891 119875

119862(119906) rarr










in 119875119862(119906)




119891x = (1199091 1198961199092 0 0 ) (36)

for all x = 119909119899119899ge1



Fix (119891) = x = 119909119899119899ge1

isin ℓ1 119909119899= 0 forall119899 ge 2 (37)





119862(x) = (1 0 0 0 ) if 119909

1gt 1

and 119875119862(x) = (minus1 0 0 0 ) if 119909


119862(119906)




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













2 Main Results







119891119899119909 = (1 minus 119896

119899) 119911 + 119896

119899119891119909 (13)


119899rarrinfin119896119899= 1 From




119865 (10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) = 119865 (

1003817100381710038171003817(1 minus 119896119899) 119911 + 119896

119899119891119909

minus119891119899((1 minus 119896

119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

= 119865 (1003817100381710038171003817(1 minus 119896

119899) 119911 + 119896

119899119891119909 minus (1 minus 119896

119899) 119911

minus119896119899119891 ((1 minus 119896

119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

= 119865 (1003817100381710038171003817119896119899 (119891119909 minus 119891 ((1 minus 119896

119899) 119911 + 119896

119899119891119909))

1003817100381710038171003817)

(14)



119865 (10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) lt 119865 (

1003817100381710038171003817119891119909 minus 119891 ((1 minus 119896119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

lt 119865 (1003817100381710038171003817119909 minus ((1 minus 119896

119899) 119911 + 119896

119899119891119909)

1003817100381710038171003817)

= 119865 (1003817100381710038171003817119909 minus 119891

1198991199091003817100381710038171003817)

(15)

Hence

119865 (1003817100381710038171003817119909 minus 119891

1198991199091003817100381710038171003817) minus 119865 (

10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) gt 0 (16)


0 lt 120591119899le 119865 (

1003817100381710038171003817119909 minus 1198911198991199091003817100381710038171003817) minus 119865 (

10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) (17)

Therefore

120591119899+ 119865 (

10038171003817100381710038171003817119891119899119909 minus 1198912

11989911990910038171003817100381710038171003817) le 119865 (

1003817100381710038171003817119909 minus 1198911198991199091003817100381710038171003817) (18)




119891119909119899119894

119894ge1




119891119909119899119894

(19)


119894rarrinfin119896119899119894

= 1 and 119909119899119894

= 119891119899119894

119909119899119894



119891119909119899119894


((1 minus 119896119899119894

) 119911 + 119896119899119894

119891119909119899119894

)


119891119899119894

119909119899119894


119909119899119894

(20)

Therefore

119891119909 = lim119894rarrinfin

119891119909119899119894

(21)

We conclude that 119909 = 119891119909






1such

that for each x = 119909119899119899ge1

isin ℓ1

x1=

infin

sum119899=1

10038161003816100381610038161199091198991003816100381610038161003816 (22)


of ℓ1will be also

denoted as (1199091 1199092 1199093 119909

119899 )


119862 = x isin ℓ1 x1le 1 (23)



119891x = (1199091 1198961199092 0 0 ) (24)


for all x = (119909119899)119899ge1



subset of R2

(119906 V) isin R2

|119906| +1003816100381610038161003816100381610038161003816

V119896

1003816100381610038161003816100381610038161003816le 1 (25)


119899119899ge1


119899= 0 whenever 119899 ge 2



Let 119865 R+


ln 2 + 119865 (119889 (119891119909 1198912

119909)) le 119865 (119889 (119909 119891119909)) (26)


119865(119889(1198910 1198911

2)) = 119865(119889(1

1

4))

= 119865(3

4) = ln(3


(27)

while

119865(119889(01

2)) = 119865(

1

2) = ln(1

2) = minus ln 2 (28)

and thus

119865(119889(01

2)) minus 119865(119889(1198910 119891

1

2)) = ln 2 minus ln 3 lt 0 (29)


120591 + 119865 (119889 (119891119909 119891119910)) le 119865 (119889 (119909 119910)) (30)


2(0) = 1



lim119899rarrinfin

119891119899


119891119899+1

(0) = 1198911 = 1 (31)




119862(119906) of best




119889 (119906 119910119899) = 119903 Since 119910

119899 119899 ge 1


subsequence 119891119910119899119894

119894ge1

of 119891119910119899119899ge1


119891119910119899119894

= 119909 Now

119865 (119903) le 119865 (119889 (119906 119909)) = 119865 ( lim119894rarrinfin

119889 (119891119906 119891119910119899119894

))


119865 (119889 (119891119906 119891119910119899119894

))


119865 (119889 (119906 119910119899119894

))


119889 (119906 119910119899119894

)) = 119865 (119903)

(32)

This implies

119865 (119903) = 119865 (119889 (119906 119909)) (33)


is nonempty







119862(119906) and set

119903 = 119889 (119906 119862) then

119865 (119903) le 119865 (119889 (119906 119891119910)) = 119865 (119889 (119891119906 119891119910))

le 119865 (119889 (119906 119910)) = 119865 (119903) (34)

This implies

119865 (119903) = 119865 (119889 (119906 119891119910)) (35)


119875119862(119906) This proves that 119875

119862(119906) is 119891-invariant so 119891 119875

119862(119906) rarr










in 119875119862(119906)




119891x = (1199091 1198961199092 0 0 ) (36)

for all x = 119909119899119899ge1



Fix (119891) = x = 119909119899119899ge1

isin ℓ1 119909119899= 0 forall119899 ge 2 (37)





119862(x) = (1 0 0 0 ) if 119909

1gt 1

and 119875119862(x) = (minus1 0 0 0 ) if 119909


119862(119906)




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










for all x = (119909119899)119899ge1



subset of R2

(119906 V) isin R2

|119906| +1003816100381610038161003816100381610038161003816

V119896

1003816100381610038161003816100381610038161003816le 1 (25)


119899119899ge1


119899= 0 whenever 119899 ge 2



Let 119865 R+


ln 2 + 119865 (119889 (119891119909 1198912

119909)) le 119865 (119889 (119909 119891119909)) (26)


119865(119889(1198910 1198911

2)) = 119865(119889(1

1

4))

= 119865(3

4) = ln(3


(27)

while

119865(119889(01

2)) = 119865(

1

2) = ln(1

2) = minus ln 2 (28)

and thus

119865(119889(01

2)) minus 119865(119889(1198910 119891

1

2)) = ln 2 minus ln 3 lt 0 (29)


120591 + 119865 (119889 (119891119909 119891119910)) le 119865 (119889 (119909 119910)) (30)


2(0) = 1



lim119899rarrinfin

119891119899


119891119899+1

(0) = 1198911 = 1 (31)




119862(119906) of best




119889 (119906 119910119899) = 119903 Since 119910

119899 119899 ge 1


subsequence 119891119910119899119894

119894ge1

of 119891119910119899119899ge1


119891119910119899119894

= 119909 Now

119865 (119903) le 119865 (119889 (119906 119909)) = 119865 ( lim119894rarrinfin

119889 (119891119906 119891119910119899119894

))


119865 (119889 (119891119906 119891119910119899119894

))


119865 (119889 (119906 119910119899119894

))


119889 (119906 119910119899119894

)) = 119865 (119903)

(32)

This implies

119865 (119903) = 119865 (119889 (119906 119909)) (33)


is nonempty







119862(119906) and set

119903 = 119889 (119906 119862) then

119865 (119903) le 119865 (119889 (119906 119891119910)) = 119865 (119889 (119891119906 119891119910))

le 119865 (119889 (119906 119910)) = 119865 (119903) (34)

This implies

119865 (119903) = 119865 (119889 (119906 119891119910)) (35)


119875119862(119906) This proves that 119875

119862(119906) is 119891-invariant so 119891 119875

119862(119906) rarr










in 119875119862(119906)




119891x = (1199091 1198961199092 0 0 ) (36)

for all x = 119909119899119899ge1



Fix (119891) = x = 119909119899119899ge1

isin ℓ1 119909119899= 0 forall119899 ge 2 (37)





119862(x) = (1 0 0 0 ) if 119909

1gt 1

and 119875119862(x) = (minus1 0 0 0 ) if 119909


119862(119906)




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













in 119875119862(119906)




119891x = (1199091 1198961199092 0 0 ) (36)

for all x = 119909119899119899ge1



Fix (119891) = x = 119909119899119899ge1

isin ℓ1 119909119899= 0 forall119899 ge 2 (37)





119862(x) = (1 0 0 0 ) if 119909

1gt 1

and 119875119862(x) = (minus1 0 0 0 ) if 119909


119862(119906)




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Generalized Contraction and Invariant Approximation …downloads.hindawi.com/journals/aaa/2014/391952.pdf · 2019. 7. 31. · Research Article Generalized Contraction