From single-variable to Pexider-type: A new direct proof for Hyers-Ulam stability of functional equations in fuzzy Banach spaces

We investigate the Hyers-Ulam stability of functional equations involving a single variable in fuzzy Banach spaces using a new direct method. This method imposes no restrictions on the domain or range of functions and is shown to be simpler and more effective for various functional equations. Furthermore, we establish a fuzzy version of the generalized Hyers-Ulam stability for a Pexider-type functional inequality and a linear functional equation with multiple coefficients in a fuzzy Banach linear space. For both equations, we obtain the existence and uniqueness of approximating solutions. To validate the theoretical results, numerical experiments are conducted using the Monte Carlo random sampling method. The results show that the mean ratio of the true error to the theoretical upper bound is only 0.0027, and the 95th percentile is 0.0051, indicating that the derived error bound is both reliable and tight. The proposed method enriches the proof techniques for stability problems of functional equations in fuzzy spaces, and the findings can serve as a reference for theoretical research and practical applications in related fields.