Viscosity Approximation to Common Fixed Points of Families of Nonexpansive Mappings with Weakly Contractive Mappings

Abstract

Let X be a reflexive Banach space which has a weakly sequentially continuous duality mapping. In this paper, we consider the following viscosity approximation sequence , where (0, 1), is a uniformly asymptotically regular sequence, and f is a weakly contractive mapping. Strong convergence of the sequence is proved.

1. Introduction

Let be a nonempty closed convex subset of a Banach space . Recall that a self-mapping is nonexpansive if

(1.1)

Alber and Guerre-Delabriere [1] defined the weakly contractive maps in Hilbert spaces, and Rhoades [2] showed that the result of [1] is also valid in the complete metric spaces as follows.

Definition 1.1.

Let be a complete metric space. A mapping is called weakly contractive if

(1.2)

where and is a continuous and nondecreasing function such that if and only if and .

Theorem 1.2.

Let be a weakly contractive mapping, where is a complete metric space, then has a unique fixed point.

In 2007, Song and Chen [3] considered the iterative sequence

(1.3)

They proved the strong convergence of the iterative sequence , where is a contraction mapping and is a uniformly asymptotically regular sequence of nonexpansive mappings in a reflexive Banach space , as follows.

Theorem 1.3 (see [3, Theorem ]).

Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping from to . Suppose that is a nonempty closed convex subset of and is a uniformly asymptotically regular sequence of nonexpansive mappings from into itself such that

(1.4)

where . Let be defined by (1.3) and , such that . Then as , the sequence converges strongly to , such that is the unique solution, in , to the variational inequality:

(1.5)

In this paper, inspired by the above results, strong convergence of sequence (1.3) is proved, where is a weakly contractive mapping.

2. Preliminaries

A Banach space is called strictly convex if

(2.1)

A Banach space is called uniformly convex, if for all there exist such that

(2.2)

The following results are well known which can be founded in [4].

(1)A uniformly convex Banach space is reflexive and strictly convex.

(2)If is a nonempty convex subset of a strictly convex Banach space and is a nonexpansive mapping, then the fixed point set of is a closed convex subset of .

By a gauge function we mean a continuous strictly increasing function defined on such that and . The mapping defined by

(2.3)

is called the duality mapping with gauge function . In the case where then which is the normalized duality mapping.

Proposition 2.1 (see [5]).

• (1) if and only if is a Hilbert space.

• (2) is surjective if and only if is reflexive.

• (3) for all ; in particular , for all .

We say that a Banach space has a weakly sequentially continuous duality mapping if there exists a gauge function such that the duality mapping is single-valued and continuous from the weak topology to the wea topology of .

We recall [6] that a Banach space is said to satisfy Opial's condition, if for any sequence in , which converges weakly to , we have

(2.4)

It is known [7] that any separable Banach space can be equivalently renormed such that it satisfies Opial's condition. A space with a weakly sequentially continuous duality mapping is easily seen to satisfy Opial's condition [8].

Lemma 2.2 (see [9, Lemma ]).

Let be a Banach space satisfying Opial's condition and a nonempty, closed, and convex subset of . Suppose that is a nonexpansive mapping. Then is demiclosed at zero, that is, if is a sequence in which converges weakly to and if the sequence converges strongly to zero, then .

Definition 2.3 (see [3]).

Let be a nonempty closed convex subset of a Banach space and , where . Then the mapping sequence is called uniformly asymptotically regular on , if for all and any bounded subset of we have

(2.5)

3. Main Result

In this section, we prove a new version of Theorem 1.3.

Theorem 3.1.

Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping from to . Suppose that is a nonempty closed convex subset of and is a uniformly asymptotically regular sequence of nonexpansive mappings such that

(3.1)

Let be a weakly contractive mapping. Suppose that is a sequence of positive numbers in satisfying . Assume that is defined by the following iterative process:

(3.2)

Then the above sequence converges strongly to a common fixed point of such that is the unique solution, in , to the variational inequality

(3.3)

Proof.

Step 1.

We prove the uniqueness of the solution to the variational inequality (3.3). Suppose that are distinct solutions to (3.3). Then

(3.4)

By adding up the above relations, we get

(3.5)

Thus , hence . We denote by the unique solution, in , to (3.3).

Step 2.

We show that the sequence is bounded. Let ; from (3.2) we get then that

(3.6)

Thus

(3.7)

or

(3.8)

Therefore is bounded.

Step 3.

We prove that , for all . Since the sequence is bounded, so and are bounded. Hence , thus . Let be a bounded subset of which contains . Since the sequence is uniformly asymptotically regular, we can obtain

(3.9)

Let , then

(3.10)

Hence , for all .

Step 4.

We show that the sequence is sequentially compact. Since is reflexive and is bounded, there exists a subsequence of such that is weakly convergent to as . Since for all , by Lemma 2.2, we have for all . Thus .

Step 2 implies that

(3.11)

Hence

(3.12)

Since is single valued and weakly sequentially continuous from to , we have

(3.13)

Thus . Hence the sequence is sequentially compact.

Step 5.

We now prove that is a solution to the variational inequality (3.3). Suppose that , then

(3.14)

Hence

(3.15)

Since as , we have

(3.16)

as . Hence

(3.17)

Thus is a solution to the variational inequality (3.3). By uniqueness, . Since the sequence is sequentially compact and each cluster point of it is equal to , then as . The proof is completed.

It is known that [10, Example ] in a uniformly convex Banach space , the Cesàro means for nonexpansive mapping is uniformly asymptotically regular. So we have the following corollary, which is a new version of [10, Theorem ].

Corollary 3.2.

Let be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from to and a nonempty closed convex subset of . Suppose that is a nonexpansive mapping, and is a weakly contractive mapping. Let be defined by

(3.18)

where and . Then as , converges strongly to a fixed point of , where is the unique solution in to the following variational inequality:

(3.19)