Contraction-Type Fixed-Point Theorem for Bivariate/Multivariate Self-Mappings in Fuzzy Banach Spaces and Hyers–Ulam Stability of Multivariate Functional Equations

To address the lack of dedicated tools for analyzing the stability of bivariate functional equations in fuzzy environments, this paper investigates fixed-point theory and functional equation stability in fuzzy Banach spaces (FBSs). First, building on the Bag–Samanta fuzzy norm, we supplement and prove the “proposition on convergence preservation of linear operations for bivariate sequences,†laying key technical groundwork. Second, we move beyond the constraints of traditional single-variable fixed-point theorems to develop a contraction-type fixed-point theorem for bivariate self-mappings in the FBS framework, clarifying criteria for the existence and uniqueness of fixed points when mappings satisfy contraction conditions. Finally, we apply this theorem to the Hyers–Ulam stability analysis of bivariate linear functional equations (e.g., generalized Jensen-type equations), proving that approximate solutions imply unique exact solutions with convergence error estimates. Our results enrich the FBS fixed-point theory system and provide a novel general method for stability analysis of multivariate functional equations under fuzzy uncertainty, with applications in dynamic system modeling and fuzzy optimization.