The first common fixed point theorem for commutative set-valued mappings

Abstract

We establish the first common fixed point theorem for commutative set-valued convex mappings. This may help to generalize common fixed point theorems in single-valued setting to those in set-valued. We also prove the existence of a fixed point in a continuously expanding sets under a none convex upper semicontinuous set-vaued mapping; as a result we answer positively to a question of Lau and Yao.