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Zbl 0826.47038
Xu, Hong-Kun; Yin, Xi-Ming
Strong convergence theorems for nonexpansive nonself-mappings.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 24, No.2, 223-228 (1995). ISSN 0362-546X

Let $C$ be a closed convex subset of a Banach space $X$, $u\in C$, and $T: C\to X$ a nonexpansive map. Then the operator $S_t$ defined for $0< t< 1$ by $S_t x= tTx+ (1- t)u$ is a contraction, and hence has a unique fixed point $x_t\in C$ if $T(C)\subseteq C$. In this paper the authors discuss various conditions under which $x_t$ converges to a fixed point of $T$ as $t\to 1$.
[J.Appell (Würzburg)]
MSC 2000:
*47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: strong convergence theorems; closed convex subset of a Banach space; nonexpansive map; contraction; unique fixed point

Cited in: Zbl 1049.47511

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