Abstract
We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let be a bounded closed convex subset of a uniformly smooth Banach space . Let be an infinite family of commuting nonexpansive mappings on . Let and be sequences in satisfying for . Fix and define a sequence in by for . Then converges strongly to , where is the unique sunny nonexpansive retraction from onto .
Highlights
Let C be a closed convex subset of a Banach space E
In this paper, using the idea in [21], we prove Browder’s type strong convergence theorems for infinite families of nonexpansive mappings without assuming the strict convexity of the Banach space
A convex subset C of a Banach space E is said to have normal structure [3] if for every bounded convex subset K of C which contains more than one point, there exists z ∈ K
Summary
Let C be a closed convex subset of a Banach space E. We denote by F(T) the set of fixed points of T. We know that F(T) is nonempty in the case that E is uniformly smooth and C is bounded; see Baillon [1]. In 1967 Browder [6] proved the following strong convergence theorem. Let C be a bounded closed convex subset of a Hilbert space E and let T be a nonexpansive mapping on C. Let {αn} be a sequence in (0, 1) converging to 0. Fix u ∈ C and define a sequence {un} in C by un = 1 − αn Tun + αnu (1.1).
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