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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)
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.
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 , proved that is nonempty provided that is, in addition, bounded. Kirk in a very celebrated paper, see , extended this result to the setting of reflexive Banach spaces with normal structure.
Browder  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 , Takahashi and Ueda , and O'Hara et al. .
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.
Theorem 1.1 (see ).
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 ,
Theorem 1.2 (see ).
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  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 .
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.
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 .
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 ).
Let be a Hilbert space. Let be a sequence in converging weakly to . Then the inequality implies .
We generalize Theorem 1.2.
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
Fix and define a sequence in by (1.2). Then converges strongly to the element of nearest to .
Put . Let be the element of nearest to . Since
we have . Therefore is bounded. Hence is also bounded.
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:
In the first case, we fix . For sufficiently large , we have
By the Opial property, we obtain . Thus .
In the second case, we have
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 .
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.
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 .
For and , we define by
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.
Let be an infinite-dimensional Hilbert space. Let and be sequences in satisfying (C1). Then the following are equivalent:
(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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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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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
- Hilbert Space
- Banach Space
- Topological Vector Space
- Type Theorem
- Implicit Method