Hyperstability of the general linear functional equation. (English) Zbl 1334.39062
For two normed spaces \(X,Y\), \(f: X\to Y\) and given scalars \(a,b,A,B\), by the general linear functional equation the author means
\[
f(ax+by)=Af(x)+Bf(y),\qquad x,y\in X.\tag{1}
\]
Its approximate solutions are defined by
\[
\|f(ax+by)-Af(x)-Bf(y)\|\leq\varphi(x,y),\qquad x,y\in X\tag{2}
\]
with some control mapping \(\varphi\). For some particular forms of \(\varphi\), the hyperstability is proved, e.g., each solution of (2) is actually a solution of (1).
Reviewer: Jacek Chmieliński (Kraków)
MSC:
39B82 | Stability, separation, extension, and related topics for functional equations |
39B52 | Functional equations for functions with more general domains and/or ranges |