We consider L- \(\omega \) -nonexpansive maps \(T:K\rightarrow K\) 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 \(\omega '(0)=\infty \) , we show that if X contains an isomorphic copy of \(\textrm{c}_0\) 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 \(\omega '(0)\le 1-L\) we show that AFPP works and FPP also works under monotonicity conditions on \(\omega \) . Further related results and examples are given.
