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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)ξ+η(x ν +y ν )

(x,yE 1 {0}) with some ξ,η0 and ν find a (possibly unique) quadratic mapping g:E 1 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 ν<2, or ν>2, ξ=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 ν=2 the quadratic functional equation is not stable in the considered sense.


MSC:
39B52Functional 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.