On $L$-$\ddot{I}‰$-nonexpansive maps

Abstract

We consider $L$-$\ddot{I}‰$-nonexpansive maps $T:K\to K$ on a convex subset $K$ of a Banach space $X$, i.e., maps in which $\ddot{I}{‰}_T(\hat{I}´)\le L\hat{I}´+\ddot{I}‰(\hat{I}´)$ with $L\in [0,1]$, $\ddot{I}‰$ being a modulus of continuity and $\ddot{I}{‰}_T$ is the minimal modulus of continuity of $T$. Both AFPP and FPP are studied. For moduli $\ddot{I}‰$ with $\ddot{I}{‰}^{\prime }(0)=\infty$, we show that if $X$ contains an isomorphic copy of $\co$ then it fails the FPP for $0$-$\ddot{I}‰$-nonexpansive maps with minimal displacement zero. In the affirmative direction, we prove for certain class of moduli $\ddot{I}‰$ that $0$-$\ddot{I}‰$-nonexpansive maps are constant on certain domains. Also, when $\ddot{I}{‰}^{\prime }(0)\le 1-L$ we show that AFPP works and FPP also works under a monotonicity condition on $\ddot{I}‰$. Further related results and examples are given.

Accepted in June 30, 2025