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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).

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges
##### Keywords:
hyperstability; general linear functional equation
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