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 : C → 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 : C → 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 of a bounded convex subset of a topological vector space has an approximate fixed point sequence. Moreover, we construct an affine sequentially continuous map from a compact convex set into itself without fixed points.