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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)
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[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
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