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Generalized stability of the Cauchy functional equation. (English) Zbl 0922.39008
The author introduces a new stability concept which generalizes the notion of the Hyers-Ulam stability. A function $$C:\mathbb R \times \mathbb R \to [-\infty,+\infty]$$ is called a comparison function and given $$\alpha \leq \beta$$ assume that a function $$f:S \to \mathbb R$$ ($$(S,+)$$ is an abelian semigroup) satisfies the following relation: $\alpha \leq C(f(x+y)-f(x),f(y))\leq \beta \qquad x,y \in S.$ Does there exist an additive function $$A:S \to \mathbb R$$ such that $\alpha \leq C(A(x),f(x))\leq \beta \qquad x\in S?$ The following theorem holds. Theorem: Let $$C$$ satisfies the following assumptions:
(i) $$C(u,u)=\gamma \in \mathbb R$$ for all $$u \in \mathbb R$$;
(ii) For each fixed $$v \in \mathbb R$$, the function $$u \mapsto C(u,v)$$ is either nondecreasing and $$\lim_{u\to \infty} C(u,v)=\infty$$ or it is nonincreasing and $$\lim_{u\to \infty} C(u,v)=-\infty$$;
(iii) For all $$-\infty<\alpha \leq \beta<\infty$$ and $$v \in \mathbb R$$, the level sets $\{u:\;C(u,v)\geq\alpha\;\} \quad \text{and} \quad \{u:\;C(u,v)\leq\beta\;\}$ are closed. If $$-\infty<\alpha \leq \gamma \leq \beta<\infty$$ and $$f:S \to \mathbb R$$ satisfies $\alpha \leq C(f(x+y)-f(x),f(y))\leq \beta \qquad x,y \in S,$ then there exists an additive function $$A:S \to \mathbb R$$ such that $\alpha \leq C(A(x),f(x))\leq \beta \qquad x\in S.$
Reviewer: G.L.Forti (Milano)

##### MSC:
 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges
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