Open Access

Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings

Fixed Point Theory and Applications20072008:469357

DOI: 10.1155/2008/469357

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

Let be a metric space and . A mapping is said to be nonexpansive if
(1.1)
and it is said to be weakly contractive if
(1.2)

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. [14] 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.

Suppose that is a metric space and , is a family of commuting self-mappings of and . The iteration of type Krasnoselski-Mann (see [7, 8]) is cyclically defined by
(1.3)
For convenience, we write
(1.4)

where the function takes values in .

Let be a closed convex subset of the normed space . Then the iteration of type Kirk (see [5, 9]) is defined by
(1.5)
Again, the iteration of type lshikawa with error (see [1012]) is defined by
(1.6)
where , , , and
(1.7)

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.

Let be a complete metric space and let be a family of commuting self-mappings, where are all nonexpansive and is weakly contractive, then there is a unique common fixed point and the iteration of type Krasnoselski-Mann generated by (1.4) converges in metric to , with the following error estimate:
(2.1)

where is the Gauss integer of .

Proof.

The uniqueness of fixed point of is clear from (1.2). Hence, the common fixed point of is unique. Let be an arbitrary point in and let be an iteration of type Krasnoselski-Mann generated by (1.4). Since is commutative, then we have . Suppose that and . Then,
(2.2)
Write for fixed . Then is a subsequence of . Since is nonexpansive and is weakly contractive, then we obtain
(2.3)
which shows , that is, is a nonincreasing sequence of nonnegative real numbers. Therefore, it tends to a limit . If , then, by nondecreasity of , Thus, from (2.3) it follows that
(2.4)
a contradiction for large enough. Therefore,
(2.5)
By (2.5), for any given , there exists such that
(2.6)
We claim that
(2.7)
In fact, from (2.6) we see that (2.7) holds when . Suppose that . If , then from (2.6) we get
(2.8)
If , then we also get
(2.9)
Therefore, by induction we derive that (2.7) holds. Since is arbitrary, is a Cauchy sequence. As is complete, we have
(2.10)
Observe that are all continuous, so is . From (2.10), it follows that
(2.11)
(2.12)

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

(2.13)
which shows
(2.14)
From (2.12), it implies that is a common fixed point of that is, . Hence, . By (2.10) and (2.14), we conclude . Set From (2.3), we have
(2.15)
Since is continuous and nondecreasing, using (2.15), it yields
(2.16)
Observe that
(2.17)

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.

Let be a Banach space and let be a nonempty closed convex set. Let be a family of commuting self-mappings, where are all nonexpansive and is weakly contractive. Then, for any , the iteration of type Kirk generated by (1.5) converges strongly to a unique common fixed point , with the following error estimate:
(2.18)

Proof.

Applying Theorem 2.1, we can suppose that is a unique common fixed point of . Since
(2.19)
we derive that is a fixed point of . Since are all nonexpansive, is weakly contractive, and , then we have
(2.20)
The inequality (2.20) shows that is weakly contractive. Thus, is a unique fixed point of . Set Then,
(2.21)
and converges to with the following error estimate (see Remark 2.2):
(2.22)
Observe that
(2.23)

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

Theorem 2.4.

Let be a Banach space and let be a nonempty closed convex set. Let be a family of commuting self-mappings, where are all nonexpansive and is weakly contractive. For any , let be the iteration of type Ishikawa generated by (1.6), where
(2.24)
and are all bounded. Then, converges strongly to a unique common fixed point with the following estimate:
(2.25)

where .

Proof.

Applying Theorem 2.1, we can suppose that is a unique common fixed point of . Since are all bounded, we have . Since are all nonexpansive and is weakly contractive, we obtain in proper order that
(2.26)
Write Then and (2.26) yields
(2.27)
(2.28)
From (2.27) and Lemma 1.1, it implies that exists, and so does by the continuity of . From (2.28), it implies that Since we conclude that Therefore, that is, converges strongly to . To establish the error estimate, we set and Then, (2.26) yields
(2.29)
Set From (2.29) we have
(2.30)
Since is nondecreasing, from (2.30) we deduce
(2.31)
Thus,
(2.32)

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

(1)
Department of Mathematics, Nanjing University of Information Science and Technology

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Copyright

© J.-Z. Xiao and X.-H. Zhu. 2008

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