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Zbl 1049.47511
Jung, Jong Soo; Kim, Seong Sik
Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces. (English)
[A] Tangmanee, E. (ed.) et al., Proceedings of the second Asian mathematical conference 1995, Nakhon Ratchasima, Thailand, October 17--20, 1995. Singapore: World Scientific. 30-33 (1998). ISBN 981-02-3225-X

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 $T$.\par 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 $T$.

MSC 2000:: *47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Citations: Zbl 0826.47038

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