The stability problem of functional equations originated from a question of S. M. Ulam [A collection of mathematical problems. New York and London: Interscience Publishers (1960; Zbl 0086.24101)] concerning the stability of group homomorphisms. D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; Zbl 0061.26403)] gave a first affirmative partial answer to the question of Ulam for Banach spaces.
The authors prove the Hyers-Ulam stability of the additive-quadratic functional equation
in non-Archimedean Banach modules over a unital Banach algebra.
The definition of non-Archimedean Banach module over a Banach algebra is not given.