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Weak convergence theorems for nonexpansive mappings and semigroups in Banach spaces with Opial’s property. (English) Zbl 0585.47043
A Banach space X satisfies Opial’s condition for the weak topology if $$x_ n\rightharpoonup y$$ in X implies $(*)\quad \limsup_{n}\| x_ n-y\| < \limsup_{n}\| x_ n-z\| \quad for\quad all\quad z\neq y.$ A Banach space X satisfies Opial’s condition for the weak-* topology if X is a conjugate space to a separable Banach space and for $$x_ n\rightharpoonup *y$$ in Y we have (*) for all $$z\neq y.$$
Let X be a Banach space with Opial’s property for the weak (weak-*) topology, C a weakly (weakly-*) compact subset of X, $$S=\{S(t):t\geq 0\}$$ be a semigroup of nonexpansive mappings on C and let $$x\in C$$. Then $$\{S(t)x\}_{t=0}$$ converges weakly (weakly-*) to a common fixed point if and only if $S(t+h)x-S(t)x\rightharpoonup 0\quad (S(t+h)x- S(t)x\rightharpoonup *0)$ as $$t\to \infty$$ for all $$h>0$$.
Reviewer: A.Smajdor

MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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