Optimal approximate fixed point results in locally convex spaces

Let CC 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)f(C) is totally bounded, then it has an approximate fixed point net. Next, it is shown that, if CC is bounded but not totally bounded, then there is a uniformly continuous map f:C→Cf: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.