Title: The first common fixed point theorem for commutative set-valued mappings
Abstract: In this paper, we establish a common fixed point theorem for commutative set-valued convex mappings. We also prove the existence of a fixed point in a continuously expanding set under a non-convex upper semicontinuous set-valued mapping. Applying this result, we develop a common fixed point theorem for a family of non-convex set-valued upper semicontinuous mappings which provides an answer to a question of Lau and Yao.