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Stability of cubic and quartic functional equations in non-Archimedean spaces. (English) Zbl 1192.39018
Using some ideas of {\it M. S. Moslehian} and {\it Th. M. Rassias} [Appl. Anal. Discrete Math. 1, No. 2, 325--334 (2007; Zbl 1257.39019)], {\it K. W. Jun} and {\it H. M. Kim} [J. Math. Anal. Appl. 274, No. 2, 867--878 (2002; Zbl 1021.39014)] and {\it W. G. Park} and {\it J. H. Bae} [Nonlinear Anal., Theory Methods Appl. 62, No. 4 (A), 643--654 (2005; Zbl 1076.39027)], the authors investigate the generalized Hyers-Ulam-Rassias stability of the cubic functional equation $$f(kx+y)+f(kx-y)=k[f(x+y)+f(x-y)]+2(k^3-k)f(x),$$ and the quartic functional equation $$f(kx+y)+f(kx-y)=k^2[f(x+y)+f(x-y)]+2k^2(k^2-1)f(x)-2(k^2-1)f(y)$$ for all $k\in \mathbb N$, where $f:G\to X$ is a mapping, $G$ is an additive group and $X$ is a complete non-Archimedean space.

39B82Stability, separation, extension, and related topics
46S10Functional analysis over fields (not $\Bbb R$, $\Bbb C$, $\Bbb H$or quaternions)
39B52Functional equations for functions with more general domains and/or ranges
Full Text: DOI arXiv
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