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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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