On $L$-$ω$-nonexpansive maps

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