Not sure whether this counts as recent enough:
Robert Cauty proved 2001 the Schauder conjecture that every continuous map of a nonempty commpact convex subset of a topological vector space (not necessarily locally convex!) has a fixed point:
- Cauty, Robert, Solution du problème de point fixe de Schauder, Fundamenta Mathematica 170, 2001, 231-246.
Although some problems have been found in the original paper, it seems that they could all be fixed.