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

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.