Generalized stability of the Cauchy functional equation
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. \textbf{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 \[ \alpha \leq C(A(x),f(x))\leq \beta \qquad x\in S. \]
scientific article; zbMATH DE number 1295519