# Common fixed-point results in uniformly convex Banach spaces

- Naknimit Akkasriworn
^{1}, - Annop Kaewkhao
^{1}Email author, - Attapol Keawkhao
^{2}and - Kritsana Sokhuma
^{3}

**2012**:171

https://doi.org/10.1186/1687-1812-2012-171

© Akkasriworn et al.; licensee Springer 2012

**Received: **27 April 2012

**Accepted: **11 September 2012

**Published: **3 October 2012

## 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 $({C}_{\lambda})$ for some $\lambda \in (0,1)$.

## Keywords

## 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 $z\in E\subseteq X$ such that $d(z,y)=dist(y,E)$ and $z=t(z)\in T(z)$, where $y\in X$, *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 $({C}_{\lambda})$ 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*, $t:E\to E$ is a single-valued quasi-nonexpansive mapping, and $T:E\to KC(E)$ is a multivalued mapping satisfying conditions $(E)$ and $({C}_{\lambda})$ for some $\lambda \in (0,1)$ such that *t* and *T* are weakly commuting, then there exists a point $z\in E$ such that $z=t(z)\in T(z)$. 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.
the fixed point set $Fix(t)$ is nonempty closed and convex, and

- 2.
for every $x\in Fix(t)$, a closed convex subset

*A*with $t(A)\subseteq A$, and $y\in A$ such that $\parallel x-y\parallel =dist(x,A)$, we have $y\in Fix(t)$.

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 $({C}_{\lambda})$ for some $\lambda \in (0,1)$. Consequently, such a theorem extends many results in the literature.

## 2 Preliminaries

In this section, we give some preliminaries.

*t*on a subset

*E*of a Banach space

*X*is called an asymptotically nonexpansive mapping if for each $n\in \mathbb{N}$, there exists a positive constant ${k}_{n}\ge 1$ with ${lim}_{n\to \mathrm{\infty}}{k}_{n}=1$ such that

If ${k}_{n}\equiv 1$ for all $n\in \mathbb{N}$, then *t* is called a nonexpansive mapping. We denote by $Fix(t)$ the set of fixed points of *t*, *i.e.*, $Fix(t)=\{x\in E:x=t(x)\}$.

*E*and by $KC(E)$ the family of nonempty compact convex subsets of

*E*. Let $H(\cdot ,\cdot )$ be the Hausdorff distance on $FB(X)$,

*i.e.*,

where $dist(a,B)=inf\{\parallel a-b\parallel :b\in B\}$ is the distance from the point *a* to the subset *B*.

*nonexpansive*if

**Definition 2.1**A multivalued mapping $T:X\to FB(X)$ is said to satisfy condition $({E}_{\mu})$ provided that

We say that *T* satisfies condition $(E)$ whenever *T* satisfies $({E}_{\mu})$ for some $\mu \ge 1$.

**Definition 2.2**A multivalued mapping $T:X\to FB(X)$ is said to satisfy condition $({C}_{\lambda})$ for some $\lambda \in (0,1)$ provided that

A point *x* is called a fixed point for a multivalued mapping *T* if $x\in T(x)$. A single valued mapping $t:E\to E$ and a multivalued mapping $T:E\to FB(E)$ are said to be commute if $t(T(x))\subseteq T(t(x))$ for all $x\in E$.

*X*is said to be strictly convex if

*X*is called uniformly convex (Clarkson [10]) if, for each $\epsilon >0$, there is a $\delta >0$ such that if $\parallel x\parallel =\parallel y\parallel =1$ then

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*$\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$, $\phi (s)=0\iff s=0$,

*such that*

*for all* $\lambda \in [0,1]$ *and all* $x,y\in X$ *such that* $\parallel x\parallel \le r$ *and* $\parallel y\parallel \le r$.

*E*be a nonempty closed and convex subset of a Banach space

*X*and $\{{x}_{n}\}$ be a bounded sequence in

*X*. For $x\in X$, define the asymptotic radius of $\{{x}_{n}\}$ at

*x*as the number

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* $0<\lambda <1$. *If*, *for every natural number* *n*, *we have* ${z}_{n+1}=\lambda {w}_{n}+(1-\lambda ){z}_{n}$ *and* $\parallel {w}_{n+1}-{w}_{n}\parallel \le \parallel {z}_{n+1}-{z}_{n}\parallel $, *then* ${lim}_{n\to \mathrm{\infty}}\parallel {w}_{n}-{z}_{n}\parallel =0$.

## 3 Main results

We recall that a mapping *t* on a subset *E* of a Banach space *X* is called quasi-nonexpansive [16] if $\parallel x-t(y)\parallel \le \parallel x-y\parallel $ for all $y\in E$ and $x\in Fix(t)$. 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* $t:E\to E$ *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 $x\in Fix(t)$ and

*A*be a closed convex subset with $t(A)\subseteq A$. Let $y\in A$ be such that $\parallel x-y\parallel =dist(x,A)$. Quasi-nonexpansiveness of

*t*implies that

Since *E* is strictly convex and $t(A)\subseteq A$, 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* $t:E\to E$ *is an asymptotically nonexpansive mapping with* $Fix(t)\ne \mathrm{\varnothing}$, *then* *t* *satisfies condition* $(K)$.

*Proof*The set $Fix(t)$ is closed and convex [[18], Theorem 2]. Let $x\in Fix(t)$,

*A*be a closed convex subset of

*E*with $t(A)\subseteq A$, and $y\in A$ be such that $\parallel x-y\parallel =dist(x,A)$. By Theorem 2.3, there exists a continuous function

*ψ*such that for all integers $l,m\ge 1$,

*A*is convex, we have

*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 ${lim}_{i\to \mathrm{\infty}}{t}^{i}(y)=z\in A$. We have

Since $z\in A$ and $\parallel x-y\parallel =dist(x,A)$, thus $y=z$. Therefore, $y\in Fix(t)$. □

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

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* $t:E\to E$ *be a mapping satisfying condition* $(K)$, *and let* $T:E\to KC(E)$ *be a multivalued mapping satisfying conditions* $(E)$ *and* $({C}_{\lambda})$ *for some* $\lambda \in (0,1)$. *If* *t* *and* *T* *are commute*, *then they have a common fixed point*, *that is*, *there exists a point* $z\in E$ *such that* $z=t(z)\in T(z)$.

*Proof* Commutative property of *t* and *T* implies that $t(T(x))\subseteq T(x)$ for all $x\in Fix(t)$. Then we have $Fix(t)\cap T(x)\ne \mathrm{\varnothing}$ for all $x\in Fix(t)$ since *t* satisfies condition $(K)$.

*T*. Take ${x}_{0}\in Fix(t)$. Since $Fix(t)\cap T({x}_{0})\ne \mathrm{\varnothing}$, we choose ${y}_{0}\in Fix(t)\cap T({x}_{0})$. Define

*t*satisfies condition $(K)$. Put

*T*satisfies condition (${C}_{\lambda}$), we have

*t*satisfies condition $(K)$, we have ${z}_{n}\in Fix(t)$. The compactness of $T(z)$ implies that the sequence $\{{z}_{n}\}$ has a convergent subsequence $\{{z}_{{n}_{k}}\}$ with the limit point $w\in T(z)$. We also obtain $w\in Fix(t)$ since $Fix(t)$ is closed. By the condition $(E)$ of

*T*, we have for some $\mu \ge 1$,

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 $z=w\in T(z)$. Hence, $z=t(z)\in T(z)$. □

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* $t:E\to E$ *be a quasi*-*nonexpansive mapping*, *and let* $T:E\to KC(E)$ *be a multivalued mapping satisfying conditions* $(E)$ *and* $({C}_{\lambda})$ *for some* $\lambda \in (0,1)$. *If* *t* *and* *T* *are commute*, *then they have a common fixed point*, *that is*, *there exists a point* $z\in E$ *such that* $z=t(z)\in T(z)$.

**Corollary 3.6** *Let* *E* *be a nonempty bounded closed convex subset of a uniformly convex Banach space* *X*. *Let* $t:E\to E$ *be an asymptotically nonexpansive mapping*, *and let* $T:E\to KC(E)$ *be a multivalued mapping satisfying conditions* $(E)$ *and* $({C}_{\lambda})$ *for some* $\lambda \in (0,1)$. *If* *t* *and* *T* *are commute*, *then they have a common fixed point*, *that is*, *there exists a point* $z\in E$ *such that* $z=t(z)\in T(z)$.

## Declarations

### 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.

## Authors’ Affiliations

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