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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 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.

$f\left(x+2y\right)+f\left(x-2y\right)+8f\left(y\right)=2f\left(x\right)+4f\left(2y\right)$
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges 46B03 Isomorphic theory (including renorming) of Banach spaces 46H25 Normed modules and Banach modules, topological modules 46S10 Functional analysis over fields (not $ℝ$, $ℂ$, $ℍ$or quaternions)