On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces

TL;DR

This paper introduces a new iterative scheme for approximating common fixed points of finite families of total asymptotically nonexpansive mappings in Banach spaces, unifying and extending existing convergence theorems.

Contribution

It presents a novel explicit iterative method that unifies previous approaches and includes new examples of mappings not previously known, with strong convergence results.

Findings

Constructed examples of total asymptotically nonexpansive mappings not asymptotically nonexpansive.

Proved strong convergence theorems for the iterative scheme in Banach spaces.

Extended and unified all known results in the literature.

Abstract

In this paper, we unify all know iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically $I$-nonexpansive mappings. Note that such a scheme contains as a particular case of the method introduced in [C.E. Chidume, E.U. Ofoedu, \textit{Inter. J. Math. & Math. Sci.} \textbf{2009}(2009) Article ID 615107, 17p]. We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically $I-$nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.