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On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings. (English) Zbl 0993.47002

Let \(\psi: \mathbb{R}_+\to \mathbb{R}_+\) be a mapping. Let \(E_1\), \(E_2\) be normed spaces. A mapping \(f: E_1\to E_2\) is said to be \(\psi\)-additive if there is a \(\theta> 0\) such that \[ \|f(x+ y)- f(x)- f(y)\|\leq\theta(\psi\|x\|+\psi\|y\|) \] for all \(x,y\in E_1\). There are given: an answer to a problem of G. Isac and Th. M. Rassias concerning Hyers-Ulam-Rassias stability of linear mappings and a new characterization of \(\psi\)-additive mappings.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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References:

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