Fixed points for uniformly asymptotically H\"older-Lipschitz mappings

Abstract

In this work, we show that if $K$ is a nonempty, closed, bounded, and convex subset of $X$ and $1\leq \gamma<\sqrt{N(X)},$ where $N(X)$ denotes the Bynum normal structure coefficient, then every uniformly asymptotically H\"older $\gamma$-Lipschitz mapping $T\colon K\to K$ 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 Jim\'enez-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 H\"older exponent from the calculations and obtain a H\"older-type extension of the Casini--Maluta, Lim--Xu, and, consequently, Kim--Xu theorems.