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Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. (English) Zbl 0686.47045

The following is the main result of this paper (Theorem 2):
Let E be a uniformly convex Banach space satisfying the Opial’s condition and C be a closed convex (but not necessarily bounded) subset of E, and T: \(C\to C\) is an asymptotically nonexpansive mapping, \(x\in C\). Then \(\{T^ nx\}\) converges weakly to a fixed point of T iff T is weakly asymptotically regular at x.
The author asks the following question: Is Theorem 2 true in all Banach spaces satisfying the Opial’s condition?
Reviewer: S.L.Singh

MSC:

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