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Stability of quartic mappings in non-Archimedean normed spaces. (English) Zbl 1179.39040
A field \(\mathcal{K}\) equipped with a function (valuation) \(|\cdot|:\mathcal{K} \to [0,\infty)\) is a non-Archimedean field, if
(i) \(|r|=0\) if and only if \(r=0\);
(ii) \(|rs|=|r|\,|s|\);
(iii) \(|r+s|\leq \max\{|r|, |s|\}\);
for all \(r, s \in \mathcal{K}\).
Similarly we define a non-Archimedean normed space. M. S. Moslehian and Th. M. Rassias [“Stability of functional equations in non-Archimedean spaces”, Appl. Anal. Discrete Math. 1, No. 2, 325–334 (2007; Zbl 1257.39019)] investigated the stability of functional equations in non-Archimedean normed spaces. The author of the present paper establishes a new method to prove the Hyers-Ulam-Rassias stability of the quartic functional equation \[ f(2x+y)+f(2x-y)+6f(y)=4[f(x+y)+f(x-y)+6f(x)] \] in non-Archimedean normed linear spaces.

39B82 Stability, separation, extension, and related topics for functional equations
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
39B52 Functional equations for functions with more general domains and/or ranges
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