A system of functional equations and its stability using fixed point method

This paper introduces and analyzes the concept of \((g,h)\)-derivations in complex Banach algebras, extending the classical notion of \(g\)-derivations. We consider a nonlinear system of three functional equations that models approximate \((g,h)\)-derivations and examine its Hyers-Ulam stability. Using a fixed point framework in generalized metric spaces, we derive the existence, uniqueness, and error bounds for the corresponding exact solutions. The results not only unify and extend previous stability results for derivations and homomorphisms but also offer a novel analytical method for treating operator equations with asymmetric structure.