Optimal approximate fixed point results in locally convex spaces

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.

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