A proof of the asymptotic conjecture

TL;DR

This paper proves a key conjecture in fixed point theory by demonstrating that under mild conditions, the minimal displacement of iterated functions dominates that of the original function, leading to fixed point existence in Banach spaces.

Contribution

It provides a new, streamlined proof of the long-standing asymptotic conjecture using minimal displacement analysis of iterated functions.

Findings

Proves the asymptotic conjecture for certain self-mappings.

Establishes fixed point existence under mild conditions in Banach spaces.

Introduces a new approach based on minimal displacement dominance.

Abstract

In this paper we prove that if $f$ is a self-mapping of a nonempty subset $K$ of a normed space $X$ that satisfies some mild conditions, then the minimal displacement of large iterations $f^n$ always dominates that of $f$ along certain $f^n$-invariant regions. As a consequence, we deduce that when $X$ is a Banach space, $K$ is closed convex and $f$ is continuous with $f^n$ being compact for some $n\geq 1$, then $f$ has at least one fixed point. This offers a new approach resulting in a streamlined proof of the long-standing asymptotic conjecture.