Optimal approximate fixed point results in locally convex spaces

TL;DR

This paper investigates conditions under which approximate fixed points exist for maps in locally convex spaces, establishing optimal results and counterexamples for various classes of maps and sets.

Contribution

It provides the first optimal approximate fixed point results in locally convex spaces, including conditions for existence and counterexamples for non-existence.

Findings

If $f(C)$ is totally bounded, an approximate fixed point net exists.

There are bounded, non-totally bounded sets with maps lacking approximate fixed points.

Every affine map on a bounded convex set has an approximate fixed point sequence.

Abstract

Let $C$ be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps $f\colon C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an approximate fixed point net. Next, it is shown that if $C$ is bounded but not totally bounded, then there is a uniformly continuous map $f\colon C\to C$ without approximate fixed point nets. We also exhibit an example of a sequentially continuous map defined on a compact convex set with no approximate fixed point sequence. In contrast, it is observed that every affine (not-necessarily continuous) self-mapping a bounded convex subset of a topological vector space has an approximate fixed point sequence. Moreover, it is constructed a affine sequentially continuous map from a compact convex set into itself without fixed points.