A New Fixed Point Theorem for Non-expansive Mappings and Its Application

TL;DR

This paper introduces a novel fixed point theorem for non-expansive mappings in Hilbert spaces using KKM theorem, with applications to integral equations and relaxed conditions compared to Banach's contraction principle.

Contribution

It presents a new fixed point theorem for non-expansive mappings leveraging KKM theorem, expanding solution methods for integral equations under weaker conditions.

Findings

Existence of fixed points for non-expansive mappings in Hilbert spaces.

Application of the theorem to solve integral equations.

Conditions weaker than Banach's contraction principle.

Abstract

We use $KKM$ theorem to prove the existence of a new fixed point theorem for non-expansive mapping:Let M be a bounded closed convex subset of Hilbert space H, and $A:M\rightarrow M$ be a non-expansive mapping, then exists a fixed point of A in M, we also apply this Theorem to study the solution for an integral equation,we can weak some conditions comparing with Banach's contraction principe.