Weak Convergence Theorems for Nonself Mappings

Abstract

Let $E$ be a real uniformly convex and smooth Banach space, and $K$ be a nonempty closed convex subset of $E$ with $P$ as a sunny nonexpansive retraction. Let $T_1, T_2 : K → E$ be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with a sequence $\{k^{(i)}_n\} ⊂ [1, ∞) (i = 1, 2)$, and $F := F(T_1) ∩ F(T_2) ≠ ∅$. An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If $E$ also has a Fréchet differentiable norm or its dual $E^∗$ has Kadec-Klee property, then weak convergence theorems are obtained.