A selection theorem for open graph correspondences with star-shaped values

Abstract

For a proof combine Theorem 1 with Brouwer Fixed Point Theorem. In the convex valued case (an infinite dimensional version of) Theorem 2 has been proved by Ky Fan (2) via the Knaster-Kuratowsky-Mazurkiewicz lemma. The need to investigate the fixed point theory of open graph correspondences has recently been felt in mathematical economics (see, for an example, Gale and Mas-Cole11 (1)). A natural approach is to reduce it to the continuous functions theory via selection theorems. This is easily accomplished in the convex valued case and Theorem I takes care of the homeomorphically convex one. It would be of interest to determine if homeomorphically convex could be weakened to contractible in the statement of Theorem 1 since as a corollary this would yield the open-graph analog of the Eilenberg-Montgomery fixed point theorem. We