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Zbl 0779.39003
Czerwik, St.
On the stability of the quadratic mapping in normed spaces.
(English)
[J] Abh. Math. Semin. Univ. Hamb. 62, 59-64 (1992). ISSN 0025-5858; ISSN 1865-8784/e

Modifying {\it 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 $E\sb 1$ into a Banach space $E\sb 2$ and satisfies the inequality $$\Vert f(x+y)+f(x-y)-2f(x)-2f(y)\Vert\le\xi+\eta(\Vert x\Vert\sp \nu+\Vert y\Vert\sp \nu)$$ $(x,y\in E\sb 1\backslash\{0\})$ with some $\xi,\eta\ge 0$ and $\nu\in\bbfR$ find a (possibly unique) quadratic mapping $g:E\sb 1\to E\sb 2$ lying not far'' from $f$.\par This approach to stability joints those of S. M. Ulam and D. H. Hyers as well as that of {\it 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 $f(0)=0$.\par A modification of {\it 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.
[W.Jarczyk (Katowice)]
MSC 2000:
*39B52 Functional equations for functions with more general domains

Citations: Zbl 0061.264; Zbl 0398.47040; Zbl 0739.39013

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