Browder's Convergence for Uniformly Asymptotically Regular Nonexpansive Semigroups in Hilbert Spaces
© G. López Acedo and T. Suzuki. 2010
Received: 6 October 2009
Accepted: 14 October 2009
Published: 15 October 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.
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. .
Theorem 1.1 (see ).
Theorem 1.2 (see ).
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:
They also showed that the conjunction of (C ) and (C ) is best possible; see also .
2. Main Results
The following proposition plays an important role in this paper.
It is well known that every Hilbert space has the Opial property.
Proposition 2.2 (Opial ).
We generalize Theorem 1.2.
By Theorem 2.3 and Example 2.5, we obtain the following.
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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