# Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings

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

**2008**:469357

**DOI: **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.

where .

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

## References

- Alber YI, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert spaces. In
*New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications*.*Volume 98*. Birkhäuser, Basel, Switzerland; 1997:7-22.View ArticleGoogle Scholar - Alber YI, Chidume CE, Zegeye A: Approximating fixed points of total asymptotically nonexpansive mappings.
*Fixed Point Theory and Applications*2006, 2006:-20.Google Scholar - Alber YI, Guerre-Delabriere S: On the projection methods for fixed point problems.
*Analysis*2001, 21(1):17-39.MATHMathSciNetView ArticleGoogle Scholar - Alber YI, Guerre-Delabriere S, Zelenko L: The principle of weakly contractive mappings in metric spaces.
*Communications on Applied Nonlinear Analysis*1998, 5(1):45-68.MATHMathSciNetGoogle Scholar - Rhoades BE: Some theorems on weakly contractive maps.
*Nonlinear Analysis: Theory, Methods and Applications*2001, 47(4):2683-2693. 10.1016/S0362-546X(01)00388-1MATHMathSciNetView ArticleGoogle Scholar - Hussain N, Khan AR: Common fixed-point results in best approximation theory.
*Applied Mathematics Letters*2003, 16(4):575-580. 10.1016/S0893-9659(03)00039-9MATHMathSciNetView ArticleGoogle Scholar - Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space.
*Journal of Mathematical Analysis and Applications*1996, 202(1):150-159. 10.1006/jmaa.1996.0308MATHMathSciNetView ArticleGoogle Scholar - Borwein J, Reich S, Shafrir I: Krasnoselski-Mann iterations in normed spaces.
*Canadian Mathematical Bulletin*1992, 35(1):21-28. 10.4153/CMB-1992-003-0MATHMathSciNetView ArticleGoogle Scholar - Kirk WA: On successive approximations for nonexpansive mappings in Banach spaces.
*Glasgow Mathematical Journal*1971, 12: 6-9. 10.1017/S0017089500001063MATHMathSciNetView ArticleGoogle Scholar - Ishikawa S: Fixed points by a new iteration method.
*Proceedings of the American Mathematical Society*1974, 44(1):147-150. 10.1090/S0002-9939-1974-0336469-5MATHMathSciNetView ArticleGoogle Scholar - Ishikawa S: Fixed points and iteration of a nonexpansive mapping in a Banach space.
*Proceedings of the American Mathematical Society*1976, 59(1):65-71. 10.1090/S0002-9939-1976-0412909-XMATHMathSciNetView ArticleGoogle Scholar - Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.
*Journal of Mathematical Analysis and Applications*1993, 178(2):301-308. 10.1006/jmaa.1993.1309MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.