On $L$-$\omega$-nonexpansive maps

TL;DR

This paper investigates fixed point properties of a class of nonexpansive maps characterized by a modulus of continuity, revealing conditions under which fixed point properties hold or fail in Banach spaces.

Contribution

It introduces and analyzes $L$-$mbda$-nonexpansive maps, establishing new fixed point results and counterexamples based on the modulus of continuity and the structure of Banach spaces.

Findings

Failure of FPP in spaces containing an isomorphic copy of c0 for certain moduli.

Existence of conditions where $L$-$mbda$-nonexpansive maps are constant.

AFPP holds when $mbda'(0)\u2264 1-L$ under monotonicity conditions.

Abstract

We consider $L$-$\omega$-nonexpansive maps $T\colon K\to K$ on a convex subset $K$ of a Banach space $X$, i.e., maps in which $\omega_T(\delta)\leq L\delta +\omega(\delta)$ with $L\in [0,1]$, $\omega$ being a modulus of continuity and $\omega_T$ is the minimal modulus of continuity of $T$. Both AFPP and FPP are studied. For moduli $\omega$ with $\omega'(0)=\infty$, we show that if $X$ contains an isomorphic copy of $\co$ then it fails the FPP for $0$-$\omega$-nonexpansive maps with minimal displacement zero. In the affirmative direction, we prove for certain class of moduli $\omega$ that $0$-$\omega$-nonexpansive maps are constant on certain domains. Also, when $\omega'(0)\leq 1-L$ we show that AFPP works and FPP also works under a monotonicity condition on $\omega$. Further related results and examples are given.