Preliminaries on pseudo-contractions in the intermediate sense for non-cyclic and cyclic self-mappings in metric spaces

TL;DR

This paper introduces a new contractive condition for extended 2-cyclic self-mappings in metric spaces, proving convergence to a unique sequence of best proximity points or fixed points under certain convexity conditions.

Contribution

It establishes a novel contractive framework for non-cyclic and cyclic self-mappings in metric spaces with convergence results.

Findings

Convergence to a unique finite sequence of best proximity points.

Existence of a unique fixed point when subsets intersect.

Applicable to uniformly convex metric spaces.

Abstract

A contractive condition is addressed for extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. It is proven that if the space is uniformly convex and the subsets are non-empty, closed and convex then all the iterations converge to a unique closed limiting finite sequence which contains the best proximity points of adjacent subsets and reduce to a unique fixed point if all such subsets intersect.