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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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##### References:
 [1] P.W. Cholewa, Remarks on the Stability of Functional Equations, Aequationes Mathematicae27 (1984), 76–86. · Zbl 0549.39006 [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 [4] D.H. Hyers, M. Ulam, Approximately Convex Functions, Proc. Amer. Math. Soc.3 (1952), 821–828. · Zbl 0047.29505 [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 [7] S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers Inc. New York 1960. · Zbl 0086.24101
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