Weak convergence of the iterations for asymptotically $G$-nonexpansive maps on Banach spaces with a graph

TL;DR

This paper proves weak convergence of iterative methods for asymptotically G-nonexpansive maps on Banach spaces with a graph, extending existing fixed point theorems and applying to locally nonexpansive maps.

Contribution

It unifies and generalizes fixed point convergence results for asymptotically G-nonexpansive maps on Banach spaces with a graph structure.

Findings

Weak convergence of iterations for asymptotically G-nonexpansive maps.

Extension of fixed point theorems to broader classes of maps.

Application to locally nonexpansive maps on special Banach spaces.

Abstract

We have derived that on certain Banach spaces having a graph structure $G$, the iterations for asymptotically $G$-nonexpansive map will converge weakly towards a fixed point. This result unifies and extends several theorems on fixed points proved by various authors for class of nonexpansive and asymptotically nonexpansive maps. As an application of this result, we derive that for maps satisfying the nonexpansive condition locally on special Banach spaces, the successive approximations converge weakly towards a fixed point.