Contractive Operators Controlled by Simulation Functions in Fuzzy Metric Spaces with Transitive $\mathcal{K}$-Closed Binary Relations

Abstract

In this study, we establish a novel fuzzy functional contraction within fuzzy metric spaces equipped with a binary relation, relying on the weaker concept of \(\mathcal{R}\)-completeness rather than the classical completeness of the entire space or its subspaces. The usual continuity requirement on the mapping is relaxed and replaced by either \(\mathcal{R}\)-continuity or the \(\mathfrak{P}\)-self-closedness of the relation’s restriction, employing a broad class of control functions \(\mathfrak{S}\). The theoretical results are illustrated with examples and an application to solving an integral equation governed by a given binary relation, accompanied by several corollaries and derived consequences. This work extends the theory of relation-theoretic fuzzy fixed points and provides a rigorous basis for further study of coincidence and common fixed points, with potential applications to nonlinear operator equations in uncertain settings.