# zbMATH — the first resource for mathematics

On the behavior of mappings which do not satisfy Hyers-Ulam stability. (English) Zbl 0761.47004
The main result of the paper is the following
Theorem. There exists a continuous function $$f:R\to R$$, satisfying $| f(x+y)-f(x)-f(y)|\leq| x|+| y|,$ for any $$x,y\in R$$, with $$\lim_{x\to\infty}(f(x)/x)=\infty$$.
This theorem gives an example to show that a stability theorem of Hyers- Rassias-Gajda-Ulam cannot be proved for $$p=1$$.

##### MSC:
 47A58 Linear operator approximation theory 41A35 Approximation by operators (in particular, by integral operators) 47J05 Equations involving nonlinear operators (general)
Full Text:
##### References:
 [1] Zbigniew Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431 – 434. · Zbl 0739.39013 · doi:10.1155/S016117129100056X · doi.org [2] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222 – 224. · Zbl 0061.26403 [3] Themistocles M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297 – 300. · Zbl 0398.47040 [4] Themistocles M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), no. 1, 106 – 113. · Zbl 0746.46038 · doi:10.1016/0022-247X(91)90270-A · doi.org [5] S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. · Zbl 0086.24101 [6] Stanislaw Ulam, Sets, numbers, and universes: selected works, The MIT Press, Cambridge, Mass.-London, 1974. Edited by W. A. Beyer, J. Mycielski and G.-C. Rota; Mathematicians of Our Time, Vol. 9. · Zbl 0558.00017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.