On Rothe’s Fixed Point Theorem in a General Topological Vector Space

Conjecture [Schauder] For every non-empty convex subset C of a topological vector space E, a compact continuous mapping f : C → C has a fixed point, i.e., a point x∗ ∈ C such that f(x∗) = x∗. (See [16], problem 54). We recall that a mapping f : C → C is said to be compact if f(C) is contained in a compact subset of C. Schauder proved in 1930 that his conjecture holds for normed vector spaces and Hukuhara proved that Schauder’s conjecture is true for locally convex spaces. In 2001, Schauder’s conjecture was resolved affirmatively by R. Cauty [2].