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

Górnicki, Jarosław

Commentat. Math. Univ. Carol. 30, No. 2, 249-252 (1989).

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.

Keywords:

asymptotic centre; uniformly convex Banach space; Opial’s condition; asymptotically nonexpansive mapping; fixed point; weakly asymptotically regular