Demiclodeness and fixed points of g-asymptotically nonexpansive mapping in Banach spaces with graph

Abstract

Let C be a nonempty closed convex subset of a uniformly convex Banach space endowed with a transitive directed graph G = (V(G), E(G)), such that V(G) =C and E(G) is convex. We introduce the definition of G-asymptotically nonexpansive self-mapping on C. It is shown that such mappings are G-demiclosed. Finally, we prove the weak and strong convergence of a sequence generated by a modified Noor iterative process to a common fixed point of a finite family of G-asymptotically nonexpansive self-mappings defined on C with nonempty common fixed points set. Our results improve and generalize several recent results in the literature.