Fixed point results for asymptotically H\"older nonexpansive type mappings

In this work, we extend Goebel-Kirk fixed point theorems to the setting of mappings of asymptotically H\"older-nonexpansive type. By providing several non-trivial examples, we show that this new framework strictly contains its classical counterparts. Furthermore, we prove that if a Banach space contains an isomorphic copy of either $c_0$ or $\ell_1$, then the fixed point property (FPP) for this class of mappings fails. Finally, we show that every infinite-dimensional Banach space contains a compact convex set $K$ admitting a fixed-point free, affine self-mapping $T$ which is of asymptotically H\"older-nonexpansive type and possesses no continuous iterates.