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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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