A Generalization of the "Brouwer-Schauder-Tychonoff" Fixed-Point Theorem

We prove a new fixed - point result for the image Im(j) of any continuous function j from K to (K x K), where K is a compact convex subset of a Hausdorff locally convex space, provided that the projection of Im(j) to the first factor is onto, and a condition on the convex hull of Im(j) holds. A special case of our result is the Brouwer - Schauder - Tychonoff fixed point theorem for continuous functions from K to K.