Fixed Point Theory and Applications

, 2008:469357 | Cite as

Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings

Open Access
Research Article

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

Let Open image in new window be a metric space and Open image in new window . A mapping Open image in new window is said to be nonexpansive if
and it is said to be weakly contractive if

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.

Suppose that Open image in new window is a metric space and Open image in new window , Open image in new window is a family of commuting self-mappings of Open image in new window and Open image in new window . The iteration Open image in new window of type Krasnoselski-Mann (see [7, 8]) is cyclically defined by
For convenience, we write

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

Let Open image in new window be a closed convex subset of the normed space Open image in new window . Then the iteration Open image in new window of type Kirk (see [5, 9]) is defined by
Again, the iteration Open image in new window of type lshikawa with error (see [10, 11, 12]) is defined by

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.

Let Open image in new window be a complete metric space and let Open image in new window be a family of commuting self-mappings, where Open image in new window are all nonexpansive and Open image in new window is weakly contractive, then there is a unique common fixed point Open image in new window and the iteration Open image in new window of type Krasnoselski-Mann generated by (1.4) converges in metric to Open image in new window , with the following error estimate:

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

Proof.

The uniqueness of fixed point of Open image in new window is clear from (1.2). Hence, the common fixed point of Open image in new window is unique. Let Open image in new window be an arbitrary point in Open image in new window and let Open image in new window be an iteration of type Krasnoselski-Mann generated by (1.4). Since Open image in new window is commutative, then we have Open image in new window . Suppose that Open image in new window and Open image in new window . Then,
Write Open image in new window for fixed Open image in new window . Then Open image in new window is a subsequence of Open image in new window . Since Open image in new window is nonexpansive and Open image in new window is weakly contractive, then we obtain
which shows Open image in new window , that is, Open image in new window is a nonincreasing sequence of nonnegative real numbers. Therefore, it tends to a limit Open image in new window . If Open image in new window , then, by nondecreasity of Open image in new window , Open image in new window Thus, from (2.3) it follows that
a contradiction for Open image in new window large enough. Therefore,
By (2.5), for any given Open image in new window , there exists Open image in new window such that
We claim that
In fact, from (2.6) we see that (2.7) holds when Open image in new window . Suppose that Open image in new window . If Open image in new window , then from (2.6) we get
Therefore, by induction we derive that (2.7) holds. Since Open image in new window is arbitrary, Open image in new window is a Cauchy sequence. As Open image in new window is complete, we have
Observe that Open image in new window are all continuous, so is Open image in new window . From (2.10), it follows that

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

which shows
From (2.12), it implies that Open image in new window is a common fixed point of Open image in new window that is, Open image in new window . Hence, Open image in new window . By (2.10) and (2.14), we conclude Open image in new window . Set Open image in new window From (2.3), we have
Since Open image in new window is continuous and nondecreasing, using (2.15), it yields
Observe that

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.

Let Open image in new window be a Banach space and let Open image in new window be a nonempty closed convex set. Let Open image in new window be a family of commuting self-mappings, where Open image in new window are all nonexpansive and Open image in new window is weakly contractive. Then, for any Open image in new window , the iteration Open image in new window of type Kirk generated by (1.5) converges strongly to a unique common fixed point Open image in new window , with the following error estimate:

Proof.

Applying Theorem 2.1, we can suppose that Open image in new window is a unique common fixed point of Open image in new window . Since
we derive that Open image in new window is a fixed point of Open image in new window . Since Open image in new window are all nonexpansive, Open image in new window is weakly contractive, and Open image in new window , then we have
The inequality (2.20) shows that Open image in new window is weakly contractive. Thus, Open image in new window is a unique fixed point of Open image in new window . Set Open image in new window Then,
and Open image in new window converges to Open image in new window with the following error estimate (see Remark 2.2):
Observe that

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

Theorem 2.4.

Let Open image in new window be a Banach space and let Open image in new window be a nonempty closed convex set. Let Open image in new window be a family of commuting self-mappings, where Open image in new window are all nonexpansive and Open image in new window is weakly contractive. For any Open image in new window , let Open image in new window be the iteration of type Ishikawa generated by (1.6), where
and Open image in new window are all bounded. Then, Open image in new window converges strongly to a unique common fixed point Open image in new window with the following estimate:

where Open image in new window .

Proof.

Applying Theorem 2.1, we can suppose that Open image in new window is a unique common fixed point of Open image in new window . Since Open image in new window are all bounded, we have Open image in new window Open image in new window . Since Open image in new window are all nonexpansive and Open image in new window is weakly contractive, we obtain in proper order that
From (2.27) and Lemma 1.1, it implies that Open image in new window exists, and so does Open image in new window by the continuity of Open image in new window . From (2.28), it implies that Open image in new window Since Open image in new window we conclude that Open image in new window Therefore, Open image in new window that is, Open image in new window converges strongly to Open image in new window . To establish the error estimate, we set Open image in new window and Open image in new window Then, (2.26) yields
Set Open image in new window From (2.29) we have
Since Open image in new window is nondecreasing, from (2.30) we deduce

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).

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Copyright information

© J.-Z. Xiao and X.-H. Zhu. 2008

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.

Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Information Science and TechnologyNanjingChina

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