Global optimization on a metric space with a graph and an application to PBVP

TL;DR

This paper introduces a new cyclic contraction mapping on metric spaces with graphs, establishing best proximity points and fixed point theorems with applications to boundary value problems, unifying many existing results.

Contribution

It presents a novel cyclic contraction mapping framework on metric spaces with graphs, linking proximity points to graph-connected substructures and applying to boundary value problems.

Findings

Number of best proximity points equals the number of connected subgraphs.

Established fixed point theorems for the new cyclic contraction.

Applied results to solve systems of boundary value problems.

Abstract

In this article we introduce a new type of cyclic contraction mapping on a pair of subsets of a metric space with a graph and prove best proximity points results for the same. Also, we demonstrate that the number of such points is same with the number of connected subgraphs. Hereafter, we introduce a fixed point mapping obtained from the aforesaid cyclic contraction and prove some fixed point theorems which will be used to find a common solution for a system of periodic boundary value problems. Our results unify and subsume many existing results in the literature.