Fixed points for uniformly asymptotically Hölder-Lipschitz mappings

In this work, we show that if $K$ is a nonempty, closed, bounded, and convex subset of $X$ and $1\le \gamma <\sqrt{N(X)},$ where $N(X)$ denotes the Bynum normal structure coefficient, then every uniformly asymptotically Hölder $\gamma$-Lipschitz mapping $T\text{\hspace{0.17em}⁣}:K\to K$ has a fixed point. To achieve this, it was necessary to guarantee the existence of a subsequence of a bounded sequence whose asymptotic diameter is controlled by the quantity $D(\{x_n\}),$ introduced by Jiménez-Melado, while simultaneously preserving the asymptotic radii at every point of the closure of the linear span generated by the original sequence. This allowed us to remove the Hölder exponent from the calculations and obtain a Hölder-type extension of the Casini–Maluta, Lim–Xu, and, consequently, Kim–Xu theorems.