Analyzing the Stability of Euler-Lagrange Additive Functional Equations in Banach Spaces: A Fixed Point and Direct Method Perspective

This study explores the Ulam-Hyers stability of the Euler-Lagrange additive functional equation in the framework of Banach spaces. By employing both direct and fixed point methods, we establish new stability results that contribute to the understanding of functional equations in normed spaces. The direct method provides explicit estimates for stability bounds, while the fixed point approach ensures the existence and uniqueness of stable solutions under suitable conditions. Our findings offer a deeper insight into the interplay between functional equations and stability theory, with potential applications in analysis and mathematical modeling.