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

We consider $L$-$\omega$-nonexpansive maps $T\text{\hspace{0.17em}⁣}:K\to K$ on a convex subset $K$ of a Banach space $X$, i.e., maps in which ${\omega }_T(\delta )\le L\delta +\omega (\delta )$ 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 ${\omega }^{\mathrm{\prime }}(0)=\mathrm{\infty }$, we show that if $X$ contains an isomorphic copy of $\text{co}$ 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 ${\omega }^{\mathrm{\prime }}(0)\le 1-L$ we show that AFPP works and FPP also works under a monotonicity condition on $\omega$. Further related results and examples are given.