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

TL;DR

This paper introduces the first common fixed point theorem for commutative set-valued mappings, extending fixed point results from single-valued to set-valued functions and addressing a question by Lau and Yao.

Contribution

It presents a novel common fixed point theorem for commutative set-valued mappings and proves fixed point existence in expanding sets under non-convex conditions.

Findings

Established the first common fixed point theorem for commutative set-valued mappings

Proved fixed point existence in non-convex, continuously expanding sets

Provided a positive answer to a question of Lau and Yao

Abstract

We establish the first common fixed point theorem for commutative set-valued 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 set under a none convex upper semicontinuous set-valued mapping; as a result we answer positivly to a question of Lau and Yao.