Open Access

Strong Convergence Theorems for Infinitely Nonexpansive Mappings in Hilbert Space

Fixed Point Theory and Applications20092009:962303

https://doi.org/10.1155/2009/962303

Received: 23 June 2009

Accepted: 12 October 2009

Published: 1 November 2009

Abstract

We introduce a modified Ishikawa iterative process for approximating a fixed point of two infinitely nonexpansive self-mappings by using the hybrid method in a Hilbert space and prove that the modified Ishikawa iterative sequence converges strongly to a common fixed point of two infinitely nonexpansive self-mappings.

1. Introduction

Let be a nonempty closed convex subset of a Hilbert space , a self-mapping of . Recall that is said to be nonexpansive if for all .

Construction of fixed points of nonexpansive mappings via Mann's iteration [1] has extensively been investigated in literature (see, e.g., [25] and reference therein). But the convergence about Mann's iteration and Ishikawa's iteration is in general not strong (see the counterexample in [6]). In order to get strong convergence, one must modify them. In 2003, Nakajo and Takahashi [7] proposed such a modification for a nonexpansive mapping .

Consider the algorithm,

(1.1)

where denotes the metric projection from onto a closed convex subset of . They prove the sequence generated by that algorithm (1.1) converges strongly to a fixed point of provided that the control sequence is chosen so that .

Let be a sequence of nonexpansive self-mappings of , a sequence of nonnegative numbers in . For each , defined a mapping of into itself as follows:

(1.2)

Such a mapping is called the -mapping generated by and ; see [8].

In this paper, motivated by [9], for any given ( is a fixed number), we will propose the following iterative progress for two infinitely nonexpansive mappings and in a Hilbert space :

(1.3)

and prove, converges strongly to a fixed point of and .

We will use the notation:

for weak convergence and for strong convergence.

denotes the weak -limit set of .

2. Preliminaries

In this paper, we need some facts and tools which are listed as lemmas below.

Lemma 2.1 (see [10]).

Let be a Hilbert space, a nonempty closed convex subset of , and a nonexpansive mapping with Fix . If is a sequence in weakly converging to and if converges strongly to , then .

Lemma 2.2 (see [11]).

Let be a nonempty bounded closed convex subset of a Hilbert space . Given also a real number and . Then the set is closed and convex.

Let be a sequence of nonexpansive self-mappings on , where is a nonempty closed convex subset of a strictly convex Banach space . Given a sequence in , one defines a sequence of self-mappings on by (1.2). Then one has the following results.

Lemma 2.3 (see [8]).

Let be a nonempty closed convex subset of a strictly convex Banach space , a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, for every and the limit exists.

Remark 2.4.

It can be known from Lemma 2.3 that if is a nonempty bounded subset of , then for there exists such that for all .

Remark 2.5.

Using Lemma 2.3, we can define a mapping as follows:
(2.1)
for all Such a is called the -mapping generated by and Since is nonexpansive mapping, is also nonexpansive. Indeed, observe that for each ,
(2.2)

If is a bounded sequence in , then we put . Hence, it is clear from Remark 2.4 that for there exists such that for all This implies that

(2.3)

Lemma 2.6 (see [8]).

Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, .

3. Strong Convergence Theorem

Theorem 3.1.

Let be a closed convex subset of a Hilbert space and let and be defined as (1.2). Assume that for all and for some and for all and . If then generated by (1.3) converges strongly to .

Proof.

Firstly, we observe that is convex by Lemma 2.2. Next, we show that for all .

Indeed, for all ,

(3.1)
Therefore,
(3.2)

That is for all . Next we show that for all .

We prove this by induction. For , we have Assume that for all since is the projection of onto so

(3.3)

As by the induction assumption, the last inequality holds, in particular, for all . This together with definition of implies that . Hence for all .

Notice that the definition of implies . This together with the fact further implies for all

The fact asserts that implies

(3.4)

We now claim that and . Indeed,

(3.5)
since , we have
(3.6)
Thus
(3.7)
We now show . Let be any subsequence of . Since is a bounded subset of , there exists a subsequence of such that
(3.8)
Since
(3.9)
it follows that . By (3.1), we have
(3.10)
Hence
(3.11)
Thus,
(3.12)
Using (3.1) again, we obtain that
(3.13)
This imply that . For the arbitrariness of , we have and
(3.14)
Thus, by (3.4), (3.7) and (3.14), we have
(3.15)
Since and we have
(3.16)
Thus, using (3.16), Lemma 2.1, and the boundedness of , we get that . Since and , we have where . By the weak lower semicontinuity of the norm, we have for all . However, since , we must have for all . Hence and
(3.17)

That is, converges to .

This completes the proof.

Declarations

Acknowledgment

This work is supported by Grant KJ080725 of the Chongqing Municipal Education Commission.

Authors’ Affiliations

(1)
College of Mathematics and Statistics, Chongqing Technology and Business University

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© Yi-An Chen. 2009

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.