Academic Journal
Generalized stability of the Cauchy functional equation
| Title: | Generalized stability of the Cauchy functional equation |
|---|---|
| Authors: | Páles, Zsolt |
| Source: | aequationes mathematicae. 56:222-232 |
| Publisher Information: | Springer Science and Business Media LLC, 1998. |
| Publication Year: | 1998 |
| Subject Terms: | Természettudományok, Systems of functional equations and inequalities, Cauchy functional equation, Abelian semigroup, Functional equations for functions with more general domains and/or ranges, Hyers-Ulam stability, Matematika- és számítástudományok, 0101 mathematics, quasi-additive functions, 01 natural sciences, separation theorem |
| Description: | 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 |
| Document Type: | Article |
| File Description: | application/xml; application/pdf |
| Language: | English |
| ISSN: | 1420-8903 0001-9054 |
| DOI: | 10.1007/s000100050058 |
| Access URL: | https://link.springer.com/article/10.1007/s000100050058 |
| Rights: | Springer TDM |
| Accession Number: | edsair.doi.dedup.....d3624bcdcb1d9c6e77d0e33c4fe4aea7 |
| Database: | OpenAIRE |
| ISSN: | 14208903 00019054 |
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| DOI: | 10.1007/s000100050058 |