# Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings

- Jian-Zhong Xiao
^{1}Email author and - Xing-Hua Zhu
^{1}

**2008**:469357

https://doi.org/10.1155/2008/469357

© J.-Z. Xiao and X.-H. Zhu. 2008

**Received: **8 August 2007

**Accepted: **26 November 2007

**Published: **12 December 2007

## Abstract

A family of commuting nonexpansive self-mappings, one of which is weakly contractive, are studied. Some convergence theorems are established for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point. The error estimates of these iterations are also given.

## 1. Introduction and Preliminaries

where is continuous and nondecreasing such that is positive on , and .

It is evident that is contractive if it is weakly contractive with , where , and it is nonexpansive if it is weakly contractive.

As an important extension of the class of contractive mappings, the class of weakly contractive mappings was introduced by Alber and Guerre-Delabriere [1]. In Hilbert and Banach spaces, Alber et al. [1–4] and Rhoades [5] established convergence theorems on iteration of fixed point for weakly contractive single mapping.

Inspired by [2, 5, 6], the purpose of this paper is to study a family of commuting nonexpansive mappings, one of which is weakly contractive, in arbitrary complete metric spaces and Banach spaces.

We will establish some convergence theorems for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point and to give their error estimates.

Throughout this paper, we assume that is the set of fixed points of a mapping , that is, ; is defined by the antiderivative (indefinite integral) of on , that is, , and is the inverse function of .

We define iterations which will be needed in the sequel.

where the function takes values in .

We will make use of following result in the proof of Theorem 2.4.

Lemma 1.1 (see [12]).

Suppose that , are two sequences of nonnegative numbers such that , for all . If , then exists.

## 2. Main Result

Theorem 2.1.

where is the Gauss integer of .

Proof.

By (1.1), (1.2), and (2.11), we deduce

From (2.16) and (2.17), we obtain the error estimate (2.1). This completes the proof.

Remark 2.2.

If in Theorem 2.1, where is the identity mapping of , then we conclude that the sequence converges to the unique common fixed point of weakly contractive mapping , with the error estimate , where . Thus, our Theorem 2.1 is a generalization of the corresponding theorem of Rhoades [5].

Theorem 2.3.

Proof.

From (2.21)–(2.23), we obtain (2.18). This completes the proof.

Theorem 2.4.

Proof.

Hence, the estimate (2.25) holds. This completes the proof.

## Declarations

### Acknowledgment

This work is supported by the National Natural Science Foundation of China (10671094).

## Authors’ Affiliations

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