Fixed point theorems for rational and trigonometric mappings in fuzzy metric spaces via fuzzy Hermite-Hadamard inequalities

This paper presents a unified framework for fixed point theory in complete fuzzy metric spaces by synthesizing fuzzy Hermite-Hadamard inequalities with various contraction types, including rational contractions. We introduce innovative fuzzy transforms \(\mathrm{M}_{\psi }\) and \(\tilde{\mathrm{M}}_{\psi }\) constructed via fuzzy Riemann integrals of convex and concave fuzzy-number-valued mappings. Our central results establish existence and uniqueness theorems under conditions involving the function class \(\Phi _{k}\) , with particular emphasis on rational-type contraction conditions that extend classical results to the fuzzy setting. The incorporation of fuzzy Hermite-Hadamard inequalities provides crucial geometric insights that enhance convergence analysis. Each theorem is accompanied by carefully constructed illustrative examples that demonstrate its applicability, and all examples are supported by graphical representations that provide visual verification of the theoretical results. The framework’s robustness is demonstrated through extensive examples encompassing both trigonometric mappings and rational contractions, revealing its adaptability to diverse operator classes. As a significant application, we establish the existence and uniqueness of fuzzy solutions to nonlinear partial differential equations, specifically fuzzy transport equations, and provide numerical examples with graphical verification of the convergence behavior. This research bridges fuzzy convex analysis with fixed point theory, offering powerful new methodologies for analyzing nonlinear operators in fuzzy environments. The integration of partial differential equations applications with graphical validation creates pathways for further applications in fuzzy variational problems and fuzzy differential equations.