  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.
