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Strong convergence theorems for nonexpansive nonself-mappings. (English) Zbl 0826.47038

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 \(t\to 1\).

MSC:

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

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