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On the stability of the quadratic mapping in normed spaces. (English) Zbl 0779.39003
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 $f$. 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 $f(0)=0$. 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.

39B52Functional equations for functions with more general domains and/or ranges
Full Text: DOI
[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
[4] D.H. Hyers, M. Ulam, Approximately Convex Functions, Proc. Amer. Math. Soc.3 (1952), 821--828. · Zbl 0047.29505 · doi:10.1090/S0002-9939-1952-0049962-5
[5] S. Kurepa, On the Quadratic Functional, Publ. Inst. Math. Acad. Serbe Sci. Beograd13 (1959), 57--72. · Zbl 0096.31501
[6] T.M. Rassias, On the Stability of the Linear Mapping in Banach Spaces, Proc. Amer. Math. Soc.72 (2) (1978), 297--300. · Zbl 0398.47040 · doi:10.1090/S0002-9939-1978-0507327-1
[7] S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers Inc. New York 1960. · Zbl 0086.24101