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Stability of a mixed additive and quadratic functional equation in non-Archimedean Banach modules. (English) Zbl 1218.39026
The stability problem of functional equations originated from a question of {\it S. M. Ulam} [A collection of mathematical problems. New York and London: Interscience Publishers (1960; Zbl 0086.24101)] concerning the stability of group homomorphisms. {\it 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 $$f(x+2y) + f(x-2y) + 8f(y) = 2f(x) + 4 f(2y)$$ in non-Archimedean Banach modules over a unital Banach algebra. The definition of non-Archimedean Banach module over a Banach algebra is not given.

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
46B03Isomorphic theory (including renorming) of Banach spaces
46H25Normed modules and Banach modules, topological modules
46S10Functional analysis over fields (not $\Bbb R$, $\Bbb C$, $\Bbb H$or quaternions)