Fixed points of asymptotically nonexpansive mappings with center 0 and applications

TL;DR

This paper studies fixed points of a special class of mappings in Banach spaces, providing new theoretical results, applications to group noncompactness, and solutions to nonlinear equations without relying on compactness.

Contribution

It introduces refined fixed point results for asymptotically nonexpansive mappings with center 0 and applies these to group theory and nonlinear equations.

Findings

Refined fixed point conditions for asymptotically nonexpansive mappings.

Characterization of noncompactness in locally compact groups.

Existence of solutions to nonlinear transport equations without compactness.

Abstract

In this paper, we investigate the existence of fixed points for asymptotically nonexpansive mappings with center 0 defined on closed convex subsets of various Banach spaces. Three applications are given. Firstly, we prove that our results refine those concerning alternate convexically nonexpansive (in short; ACN) mappings studied by P. N. Dowling in " On a fixed point result of Amini-Harandi in strictly convex Banach spaces, Acta. Math. Hungar., 112 (1-2), (2006), 85-88" . Secondly, by using Lau's result in " Closed convex invariant subsets of $L_p(G)$, Trans. Amer. Math. Soc., {\bf 232}, (1977), 131-142", we give another characterization for the noncompactness of locally compact groups $G$. Finally, we discuss the existence of a solution for a nonlinear transport equation without using compactness results.