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Stability of cubic and quartic functional equations in non-Archimedean spaces. (English) Zbl 1192.39018

Using some ideas of M. S. Moslehian and Th. M. Rassias [Appl. Anal. Discrete Math. 1, No. 2, 325–334 (2007; Zbl 1257.39019)], K. W. Jun and H. M. Kim [J. Math. Anal. Appl. 274, No. 2, 867–878 (2002; Zbl 1021.39014)] and W. G. Park and 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, where f:GX 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 , , or quaternions)
39B52Functional equations for functions with more general domains and/or ranges
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