×

On stability of the Cauchy equation in normed spaces over fields with valuation. (English) Zbl 1084.39026

The author generalizes some known results on the stability of the Cauchy equation. His main statement is the following.
Let \((X,\| \,\,\| _{1})\) be a normed space over a field \(F\) of characteristic zero with a valuation \(| \,\,| _{F},\, (Y,\| \,\, \| _{2})\) be a Banach space over a field \(K\) of characteristic zero with a valuation \(| \,\,| _{K},\, f:X\to Y\) and \(\alpha\) be a real number. If the function \(f\) satisfies \[ \| f(x+y)-f(x)-f(y)\| _{2}\leq L\max \{\| x\| ^{\alpha}_{1}, \| y\| ^{\alpha}_{1}\} \] for all \(x,y \in X\) and for some non-negative real number \(L\) and there exists a positive integer \(s\) such that \(| s| ^{\alpha}_{F}\neq | s| _{K},\) then there exist a unique additive function \(g:X\to Y\) and a real number \(C\) for which \[ \| f(x)-g(x)\| _{2}\leq C\| x\| ^{\alpha}_{1} \] for all \(x\in X.\) Here \(0^{\alpha}=0\) for \(\alpha\neq 0\) and \(0^{0}=1\).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
12J10 Valued fields
12J12 Formally \(p\)-adic fields
PDFBibTeX XMLCite