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Zbl 0761.47004
Rassias, Themistocles M.; Šemrl, Peter
On the behavior of mappings which do not satisfy Hyers-Ulam stability. (English)
[J] Proc. Am. Math. Soc. 114, No.4, 989-993 (1992). ISSN 0002-9939; ISSN 1088-6826/e

The main result of the paper is the following\par Theorem. There exists a continuous function $f:R\to R$, satisfying $$\vert f(x+y)-f(x)-f(y)\vert\leq\vert x\vert+\vert y\vert,$$ for any $x,y\in R$, with $\lim\sb{x\to\infty}(f(x)/x)=\infty$.\par This theorem gives an example to show that a stability theorem of Hyers- Rassias-Gajda-Ulam cannot be proved for $p=1$.
[A.S.Potapov (Voronezh)]

MSC 2000:: *47A58 Operator approximation theory
41A35 Approximation by operators
47J05 Equations involving nonlinear operators (general)

Keywords: approximately linear operator; stability theorem of Hyers-Rassias-Gajda- Ulam

Cited in: Zbl 0796.39012

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