fixed point results for graphical (ψ − φ)-asymptotic contractions in metric spaces with a graph

This paper is dedicated to establishing the existence and uniqueness of fixed points for g − (ψ − φ)-asymptotic contractions within the setting of metric spaces endowed with a graph. the main result presented herein constitutes a novel extension and generalization of kirk’s fixed point theorem for asymptotic contractions. specifically, we extend the theorem from classical metric spaces to the more complex framework of metric spaces equipped with a graph structure, thereby highlighting the significance and broad applicability of this class of contractions.