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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

$\parallel f\left(x+y\right)+f\left(x-y\right)-2f\left(x\right)-2f\left(y\right)\parallel \le \xi +\eta \left(\parallel x{\parallel }^{\nu }+\parallel y{\parallel }^{\nu }\right)$

$\left(x,y\in {E}_{1}\setminus \left\{0\right\}\right)$ with some $\xi ,\eta \ge 0$ and $\nu \in ℝ$ 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 $f\left(0\right)=0$.

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
##### References:
 [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). [3] D.H. Hyers, On the Stability of the Linear Functional Equation, Proc. Nat. Acad. Sci. USA27 (1941), 222–224. · doi:10.1073/pnas.27.4.222 [4] D.H. Hyers, M. Ulam, Approximately Convex Functions, Proc. Amer. Math. Soc.3 (1952), 821–828. · 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. [6] T.M. Rassias, On the Stability of the Linear Mapping in Banach Spaces, Proc. Amer. Math. Soc.72 (2) (1978), 297–300. · doi:10.1090/S0002-9939-1978-0507327-1 [7] S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers Inc. New York 1960.