# Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings

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## 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.

### Keywords

Banach Space Error Estimate Convex Subset Convergence Theorem Nonexpansive Mapping## 1. Introduction and Preliminaries

where Open image in new window is continuous and nondecreasing such that Open image in new window is positive on Open image in new window , Open image in new window and Open image in new window .

It is evident that Open image in new window is contractive if it is weakly contractive with Open image in new window , where Open image in new window , 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, 2, 3, 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 Open image in new window is the set of fixed points of a mapping Open image in new window , that is, Open image in new window ; Open image in new window is defined by the antiderivative (indefinite integral) of Open image in new window on Open image in new window , that is, Open image in new window , and Open image in new window is the inverse function of Open image in new window .

We define iterations which will be needed in the sequel.

where the Open image in new window function takes values in Open image in new window .

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

Lemma 1.1 (see [12]).

Suppose that Open image in new window , Open image in new window are two sequences of nonnegative numbers such that Open image in new window , for all Open image in new window . If Open image in new window , then Open image in new window exists.

## 2. Main Result

Theorem 2.1.

where Open image in new window is the Gauss integer of Open image in new window .

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 Open image in new window in Theorem 2.1, where Open image in new window is the identity mapping of Open image in new window , then we conclude that the sequence Open image in new window converges to the unique common fixed point Open image in new window of weakly contractive mapping Open image in new window , with the error estimate Open image in new window , where Open image in new window . 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 Open image in new window .

Proof.

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

## Notes

### Acknowledgment

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

### References

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