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

- A Razani
^{1, 2}and - S Homaeipour
^{1}Email author

**2010**:476913

https://doi.org/10.1155/2010/476913

© A. Razani and S. Homaeipour. 2010

**Received: **5 June 2010

**Accepted: **26 July 2010

**Published: **10 August 2010

## 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

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.

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.

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 ]).

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

## 2. Preliminaries

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 .

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

- (1)
- (2)
- (3)

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 .

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]).

## 3. Main Result

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

Theorem 3.1.

Proof.

Step 1.

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

Step 2.

Step 3.

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 .

Thus . Hence the sequence is sequentially compact.

Step 5.

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.

## Declarations

### Acknowledgment

A. Razani would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Teheran, Iran for supporting this paper (Grant no.89470126).

## Authors’ Affiliations

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