On pseudocontractions in cyclic maps

TL;DR

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

Contribution

It extends the theory of cyclic maps by establishing convergence results under a broader class of pseudocontractive conditions in uniformly convex metric spaces.

Findings

Iterates converge to a finite sequence of best proximity points

Unique fixed point exists if all subsets intersect

Convergence holds in uniformly convex metric spaces

Abstract

This paper discusses a more general contractive condition for a class of extended 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. If the space is uniformly convex and the subsets are non-empty, closed and convex then all the iterates converge to a unique closed limiting finite sequence which contains the best proximity points of adjacent subsets and reduces to a unique fixed point if all such subsets intersect.