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Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces. (English) Zbl 1049.47511

Tangmanee, E. (ed.) et al., Proceedings of the second Asian mathematical conference 1995, Nakhon Ratchasima, Thailand, October 17–20, 1995. Singapore: World Scientific (ISBN 981-02-3225-X). 30-33 (1998).
From the text: Theorem 1. Let \(E\) be a uniformly convex Banach space with a uniformly Gateaux differentiable norm, \(C\) a nonempty closed convex subset of \(E\), and \(T\colon C\to E\) a nonexpansive nonself-mapping. Suppose that \(C\) is a nonexpansive retract of \(E\), and that for each \(u\in C\) and \(t\in(0,1)\), the contraction \(G_t\) defined by (1) has a (unique) fixed point \(x_t\in C\). Then \(T\) has a fixed point if and only if \(\{x_t\}\) remains bounded as \(t\to 1\) and in this case, \(\{x_t\}\) converges strongly as \(t\to 1\) to a fixed point of \(T\).
Corollary 1 [H. K. Xu and X. M. Yin, Nonlinear Anal., Theory Methods Appl. 24, No.2, 223–228 (1995; Zbl 0826.47038)]. Let \(H\) be a real Hilbert space, \(C\) a nonempty closed convex subset of \(H\), and \(T\colon C\to H\) a nonexpansive nonself-mapping. Suppose that for some \(u\in C\) and each \(t\in(0,1)\), the contraction \(G_t\) defined by (1) has a (unique) fixed point \(x_t\in C\). Then \(T\) has a fixed point if and only if \(\{x_t\}\) converges strongly as \(t\to 1\) to a fixed point of \(T\).
For the entire collection see [Zbl 0937.00509].

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems

Citations:

Zbl 0826.47038
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