On a novel approach to nonexpansive mappings

TL;DR

This paper introduces perimetric nonexpansive mappings, a new class larger than existing ones, and proves fixed point existence for these mappings in Hilbert spaces, advancing the theoretical understanding of nonexpansive mappings.

Contribution

It defines a new class of nonexpansive mappings called perimetric nonexpansive mappings and proves fixed point existence results for them in Hilbert spaces.

Findings

Perimetric nonexpansive mappings form a larger class than existing mappings.

Fixed points exist for these mappings on closed bounded convex subsets of Hilbert spaces.

Connection established between periodic points and fixed points.

Abstract

This paper seeks to advance the theory of nonexpansive mappings by introducing and exploring a novel class of nonexpansive type mappings, which we aptly designate as perimetric nonexpansive mappings. We establish that the collection of mappings we propose is considerably larger than the existing classes of nonexpansive and quasi-nonexpansive mappings. We also establish fixed point existence findings by examining the connection between periodic points and fixed points in the context of normed linear spaces. Finally, we establish a significant result by proving that every perimetric nonexpansive mapping on a closed bounded convex subset of a Hilbert space necessarily has a fixed point.