×

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.
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.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
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 (if not placed in 13-XX or 16-XX)
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
PDFBibTeX XMLCite
Full Text: DOI