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

*Fixed Point Theory and Applications*
**volume 2010**, Article number: 418030 (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

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 .

Fix and define a sequence in by

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 ,

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,

for all and . Then

holds for all .

Proof.

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

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

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

We put

Let be an arbitrary subsequence of . Then there exists a subsequence of such that converges weakly to . We choose a subsequence of such that

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

(i),

(ii),

(iii).

In the first case, we fix . For sufficiently large , we have

and hence

By the Opial property, we obtain . Thus .

In the second case, we have

and hence

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

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

Hence (2.8) holds. Thus we obtain .

We next prove that converges strongly to . Since

we obtain . Since , we have

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

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

where . Define a u.a.r. nonexpansive semigroup on by

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

We note

So, . It is obvious that . We assume . Then

Arguing by contradiction, we assume . Then there exist and a subsequence of such that

Since , we have

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.

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## 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.

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### Cite this article

López Acedo, G., Suzuki, T. Browder's Convergence for Uniformly Asymptotically Regular Nonexpansive Semigroups in Hilbert Spaces.
*Fixed Point Theory Appl* **2010, **418030 (2009). https://doi.org/10.1155/2010/418030

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### Keywords

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