Fixed points for mappings of asymptotically nonexpansive type

TL;DR

This paper establishes fixed point existence for various classes of asymptotically nonexpansive mappings in Banach spaces with specific convexity or compactness properties, addressing open questions without requiring continuity.

Contribution

It proves fixed point results for asymptotically nonexpansive mappings in Banach spaces under new conditions, without assuming continuity of the mappings.

Findings

Fixed points exist for pointwise asymptotically nonexpansive mappings in certain Banach spaces.

Addresses open problems by removing the need for continuous iterates.

Results apply to nearly uniformly convex spaces and spaces with compact asymptotic centers.

Abstract

We prove the existence of a fixed point for mappings which satisfy some asymptotic nonexpansive conditions in Banach spaces which are either nearly uniformly convex or they satisfy that asymptotic centers of bounded sequences are compact. Nominally, we consider pointwise eventually nonexpansive mappings, pointwise asymptotically nonexpansive mappings and asymptotically type nonexpansive. We do not assume the existence of a continuous iterated, solving some long-standing open questions about existence of a fixed point for these mappings in absence of continuity.