- Research Article
- Open Access

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

- Genaro López Acedo
^{1}Email author and - Tomonari Suzuki
^{2}

**2010**:418030

https://doi.org/10.1155/2010/418030

© G. López Acedo and T. Suzuki. 2010

**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.

## Keywords

- Hilbert Space
- Banach Space
- Topological Vector Space
- Type Theorem
- Implicit Method

## 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.

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, [7–13]. 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, 14–18] 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 .

Then converges strongly to the element of nearest to .

We note that (C1) is needed to define .

*uniformly asymptotically regular*(

*u.a.r.*) if for every and for every bounded subset of ,

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.

holds for all .

Proof.

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.

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

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

(i) ,

(ii) ,

(iii) .

By the Opial property, we obtain . Thus .

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

Hence (2.8) holds. Thus we obtain .

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.

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.

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.

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

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