Stability of a General Functional Equation

In this paper, we investigate a general multivariable functional equation. We prove, using the fixed-point method, the generalized Hyers–Ulam stability of this equation in Banach spaces. In this way, we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. We also show, using examples, how some additional assumptions imposed on the function when examining the Hyers–Ulam stability of a functional equation affect the size of the approximating constant and limit the number of considered solutions for this equation. The functional equation studied in this paper has symmetric coefficients (with precision up to the sign), and it is a generalization of an equation characterizing n-quadratic functions, as well as many other functional equations with symmetric coefficients: for example, the multi-Cauchy equation and the multi-Jensen equation. Our results generalize many known outcomes.