Open Access

Browder's Convergence for Uniformly Asymptotically Regular Nonexpansive Semigroups in Hilbert Spaces

Fixed Point Theory and Applications20092010:418030

DOI: 10.1155/2010/418030

Received: 6 October 2009

Accepted: 14 October 2009

Published: 15 October 2009

Abstract

We give a sufficient and necessary condition concerning a Browder's convergence type theorem for uniformly asymptotically regular one-parameter nonexpansive semigroups in Hilbert spaces.

1. Introduction

Let be a closed convex subset of a Hilbert space . A mapping on is called a nonexpansive mapping if for all . We denote by the set of fixed points of . Browder, see [1], proved that is nonempty provided that is, in addition, bounded. Kirk in a very celebrated paper, see [2], extended this result to the setting of reflexive Banach spaces with normal structure.

Browder [3] initiated the investigation of an implicit method for approximating fixed points of nonexpansive self-mappings defined on a Hilbert space. Fix , he studied the implicit iterative algorithm
(1.1)

Namely, , , is the unique fixed point of the contraction , . Browder proved that , where is the element of nearest to . Extensions to the framework of Banach spaces of Browder's convergence results have been done by many authors, including Reich [4], Takahashi and Ueda [5], and O'Hara et al. [6].

A family of mappings is called a one-parameter strongly continuous semigroup of nonexpansive mappings (nonexpansive semigroup, for short) on if the following are satisfied.

(NS1) For each , is a nonexpansive mapping on .

(NS2) for all .

(NS3) For each , the mapping from into is strongly continuous.

There are many papers concerning the existence of common fixed points of ; see, for instance, [713]. As a matter of fact, Browder [8] proved that if is bounded, then is nonempty.

Browder's type convergence theorem for nonexpansive semigroups is proved in [11, 1418] and others. For example, the following theorem is proved in [17].

Theorem 1.1 (see [17]).

Let be a closed convex subset of a Hilbert space . Let be a nonexpansive semigroup on such that . Let and be sequences in satisfying

(C1) and ;

(C2) , where .

Fix and define a sequence in by
(1.2)

Then converges strongly to the element of nearest to .

We note that (C1) is needed to define .

A nonexpansive semigroup on is said to be uniformly asymptotically regular (u.a.r.) if for every and for every bounded subset of ,
(1.3)

holds. The following is proved by Domínguez Benavides et al. [16]; see also [15].

Theorem 1.2 (see [16]).

Let , and be as in Theorem 1.1. Assume that is u.a.r. Let and be sequences in satisfying (C1) and

(D2) and .

Fix and define a sequence in by (1.2). Then converges strongly to the element of nearest to .

There is an interesting difference between Theorems 1.1 and 1.2, that is, in Theorem 1.1 converges to and in Theorem 1.2 diverges to . By the way, very recently, Akiyama and Suzuki [14] generalized Theorem 1.1. They replaced (C2) of Theorem 1.1 by the following:

(C) is bounded;

(C) for all .

They also showed that the conjunction of (C ) and (C ) is best possible; see also [18].

In this paper, motivated by the previous considerations, we generalize Theorem 1.2 concerning and . Also, we will show that our new condition is best possible.

2. Main Results

We denote by the set of all positive integers and by the set of all real numbers. For , we denote by the maximum integer not exceeding .

The following proposition plays an important role in this paper.

Proposition 2.1.

Let be a set of a separated topological vector space . Let be a family of mappings on such that for all . Assume that is asymptotic regular, that is,
(2.1)
for all and . Then
(2.2)

holds for all .

Proof.

Fix . It is obvious that holds. Let be a fixed point of . For every , we have
(2.3)

and hence is a common fixed point of .

It is well known that every Hilbert space has the Opial property.

Proposition 2.2 (Opial [19]).

Let be a Hilbert space. Let be a sequence in converging weakly to . Then the inequality implies .

We generalize Theorem 1.2.

Theorem 2.3.

Let be a closed convex subset of a Hilbert space . Let be a u.a.r. nonexpansive semigroup on such that . Let and be sequences in satisfying (C1) and

(D) .

Fix and define a sequence in by (1.2). Then converges strongly to the element of nearest to .

Proof.

Put . Let be the element of nearest to . Since
(2.4)

we have . Therefore is bounded. Hence is also bounded.

We put
(2.5)
Let be an arbitrary subsequence of . Then there exists a subsequence of such that converges weakly to . We choose a subsequence of such that
(2.6)

Put , and . We will show , dividing the following three cases:

(i) ,

(ii) ,

(iii) .

In the first case, we fix . For sufficiently large , we have
(2.7)
and hence
(2.8)

By the Opial property, we obtain . Thus .

In the second case, we have
(2.9)
and hence
(2.10)

By the Opial property, we obtain . By Proposition 2.1, we obtain .

In the third case, we fix . For sufficiently large , we have
(2.11)

Hence (2.8) holds. Thus we obtain .

We next prove that converges strongly to . Since
(2.12)
we obtain . Since , we have
(2.13)

and hence converges strongly to . Since is arbitrary, we obtain that converges strongly to .

Using [20, Theorem  7], we obtain the following Moudafi's type convergence theorem; see [21].

Corollary 2.4.

Let , , , and be as in Theorem 2.3. Let be a contraction on ; that is, there exists such that for . Define a sequence in by
(2.14)

Then converges strongly to the unique point satisfying , where is the metric projection from onto .

We will show that (D ) is best possible.

Example 2.5.

Put , that is, is a Hilbert space consisting of all the functions from into satisfying with inner product . Define a bounded closed convex subset of by
(2.15)
where . Define a u.a.r. nonexpansive semigroup on by
(2.16)

Let be the canonical basis of and put . Let and be sequences in satisfying (C1) and define in by (1.2). Then converges to a common fixed point of only if .

Proof.

For and , we define by
(2.17)
We note
(2.18)
So, . It is obvious that . We assume . Then
(2.19)
Arguing by contradiction, we assume . Then there exist and a subsequence of such that
(2.20)
Since , we have
(2.21)

which is a contradiction. Therefore we obtain .

By Theorem 2.3 and Example 2.5, we obtain the following.

Theorem 2.6.

Let be an infinite-dimensional Hilbert space. Let and be sequences in satisfying (C1). Then the following are equivalent:

(i) ,

(ii)if is a bounded closed convex subset of , is a u.a.r. nonexpansive semigroup on , and is a sequence in defined by (1.2), then converges strongly to the element of nearest to .

Compare (D ) with the conjunction of (C ) and (C ). We can tell that the difference between both conditions is u.a.r.

Declarations

Acknowledgments

The first author was partially supported by DGES, Grant MTM2006-13997-C02-01 and Junta de Andalucía, Grant FQM-127. The second author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.

Authors’ Affiliations

(1)
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla
(2)
Department of Mathematics, Kyushu Institute of Technology, Tobata

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Copyright

© G. López Acedo and T. Suzuki. 2010

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.