Contractive Operators Controlled by Simulation Functions in Fuzzy Metric Spaces with Transitive K-Closed Binary Relations

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 \(R\)-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 \(R\)-continuity or the \(P\)-self-closedness of the relation’s restriction, employing a broad class of control functions \(S\). 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.