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Common fixed-point results in uniformly convex Banach spaces

Abstract

We introduce a condition on mappings, namely condition (K). In a uniformly convex Banach space, the condition is weaker than quasi-nonexpansiveness and weaker than asymptotic nonexpansiveness. We also present the existence theorem of common fixed points for a commuting pair consisting of a mapping satisfying condition (K) and a multivalued mapping satisfying conditions (E) and ( C λ ) for some λ(0,1).

1 Introduction

In 1978, Itoh and Takahashi [1] established the existence of common fixed points of a quasi-nonexpansive mapping and a multivalued nonexpansive mapping by an elementary constructive method in a Hilbert space. In 2006, Dhompongsa et al. [2] obtained a common fixed point result for a commuting pair of single-valued and multivalued nonexpansive mappings in uniformly convex Banach spaces. The analogy result in CAT(0) spaces was also proved by Dhompongsa et al. [3]. Since then, Shahzad and Markin [4] studied an invariant approximation problem and provided sufficient conditions for the existence of zEX such that d(z,y)=dist(y,E) and z=t(z)T(z), where yX, t and T are commuting nonexpansive mappings on E. In 2009, Shahzad [5] also obtained a common fixed point and invariant approximation result in a CAT(0) space in which t and T are weakly commuting.

Motivated by Suzuki’s result [6], Garcia-Falset et al. [7] introduced two kinds of generalizations for condition (C), namely conditions (E) and ( C λ ) and studied both the existence of fixed points and their asymptotic behavior. Recently, Abkar and Eslamian [8] proved that if E is a nonempty closed convex and bounded subset of a complete CAT(0) space X, t:EE is a single-valued quasi-nonexpansive mapping, and T:EKC(E) is a multivalued mapping satisfying conditions (E) and ( C λ ) for some λ(0,1) such that t and T are weakly commuting, then there exists a point zE such that z=t(z)T(z). This result was extended to the general setting of uniformly convex metric spaces by Laowang and Panyanak [9].

In this paper, we first introduce the following condition.

Definition 1.1 Let t be a mapping on a subset E of a Banach space X. Then t is said to satisfy condition (K) if

  1. 1.

    the fixed point set Fix(t) is nonempty closed and convex, and

  2. 2.

    for every xFix(t), a closed convex subset A with t(A)A, and yA such that xy=dist(x,A), we have yFix(t).

We show that, in a uniformly convex Banach space, condition (K) is weaker than quasi-nonexpansiveness and weaker than asymptotic nonexpansiveness. We also present the existence theorem of common fixed points for a commuting pair consisting of a mapping satisfying condition (K) and a multivalued mapping satisfying conditions (E) and ( C λ ) for some λ(0,1). Consequently, such a theorem extends many results in the literature.

2 Preliminaries

In this section, we give some preliminaries.

A mapping t on a subset E of a Banach space X is called an asymptotically nonexpansive mapping if for each nN, there exists a positive constant k n 1 with lim n k n =1 such that

t n ( x ) t n ( y ) k n xyfor all x,yE.

If k n 1 for all nN, then t is called a nonexpansive mapping. We denote by Fix(t) the set of fixed points of t, i.e., Fix(t)={xE:x=t(x)}.

We shall denote by FB(E) the family of nonempty bounded closed subsets of E and by KC(E) the family of nonempty compact convex subsets of E. Let H(,) be the Hausdorff distance on FB(X), i.e.,

H(A,B)=max { sup a A dist ( a , B ) , sup b B dist ( b , A ) } ,A,BFB(X),

where dist(a,B)=inf{ab:bB} is the distance from the point a to the subset B.

A multivalued mapping T:EFB(X) is said to be nonexpansive if

H ( T ( x ) , T ( y ) ) xyfor all x,yE.

Definition 2.1 A multivalued mapping T:XFB(X) is said to satisfy condition ( E μ ) provided that

dist ( x , T ( y ) ) μdist ( x , T ( x ) ) +xy,x,yX.

We say that T satisfies condition (E) whenever T satisfies ( E μ ) for some μ1.

Definition 2.2 A multivalued mapping T:XFB(X) is said to satisfy condition ( C λ ) for some λ(0,1) provided that

λdist ( x , T ( x ) ) xyH ( T ( x ) , T ( y ) ) xy,x,yX.

A point x is called a fixed point for a multivalued mapping T if xT(x). A single valued mapping t:EE and a multivalued mapping T:EFB(E) are said to be commute if t(T(x))T(t(x)) for all xE.

A Banach space X is said to be strictly convex if

x+y<2

for all x,yX with x=y=1 and xy. We recall that a Banach space X is called uniformly convex (Clarkson [10]) if, for each ε>0, there is a δ>0 such that if x=y=1 then

( x + y ) 2 1δ.

It is obvious that uniform convexity implies strict convexity.

In 1991, Xu [11] proved the characterization of uniform convexity as follows.

Theorem 2.3 [11]

A Banach space X is uniformly convex if and only if for each fixed number r>0, there exists a continuous function φ:[0,)[0,), φ(s)=0s=0, such that

λ x + ( 1 λ ) y 2 λ x 2 +(1λ) y 2 λ(1λ)φ ( x y )

for all λ[0,1] and all x,yX such that xr and yr.

Let E be a nonempty closed and convex subset of a Banach space X and { x n } be a bounded sequence in X. For xX, define the asymptotic radius of { x n } at x as the number

r ( x , { x n } ) = lim sup n x n x.

Let

rr ( E , { x n } ) :=inf { r ( x , { x n } ) : x E }

and

AA ( E , { x n } ) := { x E : r ( x , { x n } ) = r } .

The number r and the set A are, respectively, called the asymptotic radius and asymptotic center of { x n } relative to E. It is known that A(E,{ x n }) is as nonempty, weakly compact and convex as E is [12]. The sequence { x n } is called regular relative to E if r(E,{ x n })=r(E,{ x n k }) for each subsequence { x n k } of { x n }.

Goebel [13] and Lim [14] proved the following lemma.

Lemma 2.4 Let { x n } be a bounded sequence in X, and let E be a nonempty closed convex subset of X. Then { x n } has a subsequence which is regular relative to E.

The following result was proved by Goebel and Kirk [15].

Lemma 2.5 Let { z n } and { w n } be bounded sequences in a Banach space X, and let 0<λ<1. If, for every natural number n, we have z n + 1 =λ w n +(1λ) z n and w n + 1 w n z n + 1 z n , then lim n w n z n =0.

3 Main results

We recall that a mapping t on a subset E of a Banach space X is called quasi-nonexpansive [16] if xt(y)xy for all yE and xFix(t). From the definition, we can see that nonexpansive mappings with a fixed point are quasi-nonexpansive.

We first show that a quasi-nonexpansive mapping defined on a strictly convex Banach space X satisfies condition (K).

Proposition 3.1 Let E be a strictly convex subset of a Banach space. If t:EE is a quasi-nonexpansive mapping, then t satisfies condition (K).

Proof It is known that Fix(t) is nonempty closed and convex [[17], Theorem 1]. Let xFix(t) and A be a closed convex subset with t(A)A. Let yA be such that xy=dist(x,A). Quasi-nonexpansiveness of t implies that

x t ( y ) xy.

Since E is strictly convex and t(A)A, it must be the case that t(y)=y. Therefore, t satisfies condition (K). □

An asymptotically nonexpansive mapping defined on a uniformly convex Banach space also satisfies condition (K).

Proposition 3.2 Let X be a uniformly convex Banach space and E be a nonempty subset of X. If t:EE is an asymptotically nonexpansive mapping with Fix(t), then t satisfies condition (K).

Proof The set Fix(t) is closed and convex [[18], Theorem 2]. Let xFix(t), A be a closed convex subset of E with t(A)A, and yA be such that xy=dist(x,A). By Theorem 2.3, there exists a continuous function ψ such that for all integers l,m1,

x ( t l ( y ) + t m ( y ) 2 ) 2 1 2 x t l ( y ) 2 + 1 2 x t m ( y ) 2 1 4 φ ( t l ( y ) t m ( y ) ) 1 2 k l 2 x y 2 + 1 2 k m 2 x y 2 1 4 φ ( t l ( y ) t m ( y ) ) .
(3.1)

Since xy=dist(x,A) and A is convex, we have

x y 2 x ( t l ( y ) + t m ( y ) 2 ) 2 .

Thus,

φ ( t l ( y ) t m ( y ) ) 4 ( ( k l 2 + k m 2 ) / 2 1 ) x y 2 .

Since t is asymptotically nonexpansive, the right-hand side of the inequality tends to zero as l, m tend to infinity. Hence, { t i (y)} is a Cauchy sequence. Let lim i t i (y)=zA. We have

t ( z ) t i + 1 ( y ) k 1 z t i ( y ) .

By letting i, we can conclude that t(z)z0, that is zFix(t). Now, letting l,m in (3.1) yields

x z 2 x y 2 .

Since zA and xy=dist(x,A), thus y=z. Therefore, yFix(t). □

The following example shows that the class of mappings satisfying condition (K) is strictly wider than the class of quasi-nonexpansive mappings and asymptotically nonexpansive mappings.

Example 3.3 Let f be a function on [0,1] defined by

f(x)={ x , x 1 2 ; 0 , x > 1 2 .

Then f is neither quasi-nonexpansive nor asymptotically nonexpansive. However, f satisfies condition (K).

We are now in a position to state our main theorem.

Theorem 3.4 Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let t:EE be a mapping satisfying condition (K), and let T:EKC(E) be a multivalued mapping satisfying conditions (E) and ( C λ ) for some λ(0,1). If t and T are commute, then they have a common fixed point, that is, there exists a point zE such that z=t(z)T(z).

Proof Commutative property of t and T implies that t(T(x))T(x) for all xFix(t). Then we have Fix(t)T(x) for all xFix(t) since t satisfies condition (K).

Now, we find an approximate fixed point sequence in Fix(t) for T. Take x 0 Fix(t). Since Fix(t)T( x 0 ), we choose y 0 Fix(t)T( x 0 ). Define

x 1 =(1λ) x 0 +λ y 0 .

Since Fix(t) is convex, we have x 1 Fix(t). Let y 1 T( x 1 ) such that y 0 y 1 =dist( y 0 ,T( x 1 )). We get y 1 Fix(t) since t satisfies condition (K). Put

x 2 =(1λ) x 1 +λ y 1 .

Again, choose y 2 T( x 2 ) such that y 1 y 2 =dist( y 1 ,T( x 2 )). Similarly, we get y 2 Fix(t). We have a sequence { x n }Fix(t) such that

x n + 1 =(1λ) x n +λ y n ,

where y n T( x n )Fix(t) and y n 1 y n =dist( y n 1 ,T( x n )). For every natural number n1, we have

x n + 1 x n =λ x n y n .

It follows that

λdist ( x n , T ( x n ) ) λ x n y n = x n + 1 x n .

Since T satisfies condition ( C λ ), we have

H ( T ( x n ) , T ( x n + 1 ) ) x n x n + 1 .

Hence, for each n1, we have

y n y n + 1 =dist ( y n , T ( x n + 1 ) ) H ( T ( x n ) , T ( x n + 1 ) ) x n x n + 1 .

We now apply Lemma 2.5 to conclude that lim n x n y n =0. That is, as n,

dist ( x n , T ( x n ) ) x n y n 0.

Passing through a subsequence, if necessary, we can assume that { x n } is regular. Let A(Fix(t),{ x n })={z}. For each n1, we choose z n T(z) such that x n z n =dist( x n ,T(z)). Since t satisfies condition (K), we have z n Fix(t). The compactness of T(z) implies that the sequence { z n } has a convergent subsequence { z n k } with the limit point wT(z). We also obtain wFix(t) since Fix(t) is closed. By the condition (E) of T, we have for some μ1,

dist ( x n k , T ( z ) ) μdist ( x n k , T ( x n k ) ) + x n k z.

Note that

x n k w x n k z n k + z n k wμdist ( x n k , T ( x n k ) ) + x n k z+ z n k w.

These entail

lim sup k x n k w lim sup k x n k z.

Since { x n } is regular, and an asymptotic center of a bounded sequence in a uniformly convex Banach space is a singleton set, these show that z=wT(z). Hence, z=t(z)T(z). □

As a consequence of Proposition 3.1, Proposition 3.2, and Theorem 3.4, we obtain the following corollaries.

Corollary 3.5 Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let t:EE be a quasi-nonexpansive mapping, and let T:EKC(E) be a multivalued mapping satisfying conditions (E) and ( C λ ) for some λ(0,1). If t and T are commute, then they have a common fixed point, that is, there exists a point zE such that z=t(z)T(z).

Corollary 3.6 Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let t:EE be an asymptotically nonexpansive mapping, and let T:EKC(E) be a multivalued mapping satisfying conditions (E) and ( C λ ) for some λ(0,1). If t and T are commute, then they have a common fixed point, that is, there exists a point zE such that z=t(z)T(z).

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Acknowledgements

The first author would like to thank the Office of the Higher Education Commission, Thailand, for supporting grant fund under the program of Strategic Scholarships for Frontier Research Network for the PhD Program, Thai Doctoral degree, for this research.

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Correspondence to Annop Kaewkhao.

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Akkasriworn, N., Kaewkhao, A., Keawkhao, A. et al. Common fixed-point results in uniformly convex Banach spaces. Fixed Point Theory Appl 2012, 171 (2012). https://doi.org/10.1186/1687-1812-2012-171

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Keywords

  • common fixed point
  • quasi-nonexpansive mapping
  • generalized nonexpansive mapping
  • uniformly convex Banach space