A new contraction and existence theorems on fuzzy metric space with a graph

Abstract

In the present paper our aim is to develop coupled fixed point theorems in fuzzy metric space with graph.  We introduce the concept of \(\mathcal{J}-γ\)-contraction mapping using the control function developed by Wardowski [16].  In current paper, we show the existence of coupled coincidence fixed point in fuzzy metric space with respect to graph.  We also give the result having particular value of control function such that the \(\mathcal{J}-γ\)-contraction change to\(\mathcal{J}\)-fuzzy contraction.