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

Zbl 0826.47038