The author formulates, in a general form, the method of proving the Hyers-Ulam stability for functional equations in several variables. This method appeared in his paper [Stochastica 4, No. 1, 23–30 (1980; Zbl 0442.39005)] and has been actually repeated in numerous papers of various authors.
The main result reads as follows: Assume that is a set, a complete metric space and , given functions. Let satisfy the inequality
for some function . If is continuous and satisfies:
for a non-decreasing subadditive function , and the series is convergent for every , then there exists a unique function – a solution of the functional equation
Moreover, an analogous result, for a mapping satisfying the inequality