arXiv:2112.04385v1 [math.FA] 8 Dec 2021 JOTA manuscript No. (will be inserted by the editor) Global optimization on a metric space with a graph and an application to PBVP Abhik Digar · G. Sankara Raju Kosuru Received: date / Accepted: date Abstract In this article we introduce a new type of cyclic contraction map- ping on a pair of subsets of a metric space with a graph and prove best prox- imity 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 contrac- tion 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 Keywords best proximity point · fixed point · G-cyclic contraction · metric space with graph · periodic boundary value problem Mathematics Subject Classification (2010) 47H10 · 34B15 · 54H25 Department of Mathematics Indian Institute of Technology Ropar Rupnagar - 140 001, Punjab, India. abhikdigar@gmail.com · raju@iitrpr.ac.in