zbMATH — the first resource for mathematics

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

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