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On the stability of the quadratic mapping in normed spaces. (English) Zbl 0779.39003
Modifying D. H. Hyers’ classical method of studying approximately additive functions [Proc. Nat. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.264)] the author proves two results concerning the following stability problem for quadratic mappings: Assuming that \(f\) maps a normed space \(E_ 1\) into a Banach space \(E_ 2\) and satisfies the inequality \[ \| f(x+y)+f(x-y)-2f(x)-2f(y)\|\leq\xi+\eta(\| x\|^ \nu+\| y\|^ \nu) \] \((x,y\in E_ 1\backslash\{0\})\) with some \(\xi,\eta\geq 0\) and \(\nu\in\mathbb{R}\) find a (possibly unique) quadratic mapping \(g:E_ 1\to E_ 2\) lying “not far” from \(f\).
This approach to stability joints those of S. M. Ulam and D. H. Hyers as well as that of T. M. Rassias [Proc. Amer. Math. Soc. 72, 297-300 (1978; Zbl 0398.47040)]. The author answers the question in the affirmative in the case where either \(\nu<2\), or \(\nu>2\), \(\xi=0\) and \(f(0)=0\).
A modification of Z. Gajda’s example [Internat. J. Math. Math. Sci. 14, No. 3, 431-434 (1991; Zbl 0739.39013)] shows that in the critical case \(\nu=2\) the quadratic functional equation is not stable in the considered sense.

MSC:
39B52 Functional equations for functions with more general domains and/or ranges
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[1] P.W. Cholewa, Remarks on the Stability of Functional Equations, Aequationes Mathematicae27 (1984), 76–86. · Zbl 0549.39006 · doi:10.1007/BF02192660
[2] Z. Gajda, On the Stability of the Linear Mapping (to appear). · Zbl 0658.39006
[3] D.H. Hyers, On the Stability of the Linear Functional Equation, Proc. Nat. Acad. Sci. USA27 (1941), 222–224. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
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[7] S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers Inc. New York 1960. · Zbl 0086.24101
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