On the stability of the quadratic mapping in normed spaces. (English) Zbl 0779.39003

Czerwik, St.

Abh. Math. Semin. Univ. Hamb. 62, 59-64 (1992).

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.

Reviewer: W.Jarczyk (Katowice)

MSC:

39B52

Functional equations for functions with more general domains and/or ranges

Keywords:

approximately additive functions; stability; quadratic mappings; quadratic functional equation

Citations:

Zbl 0061.264; Zbl 0398.47040; Zbl 0739.39013

References:

[1]	Cholewa, P. W., Remarks on the Stability of Functional Equations, Aequationes Mathematicae, 27, 76-86 (1984) · Zbl 0549.39006
[2]	Z. Gajda, On the Stability of the Linear Mapping (to appear). · Zbl 0658.39006
[3]	Hyers, D. H., On the Stability of the Linear Functional Equation, Proc. Nat. Acad. Sci. USA, 27, 222-224 (1941) · JFM 67.0424.01
[4]	Hyers, D. H.; Ulam, M., Approximately Convex Functions, Proc. Amer. Math. Soc., 3, 821-828 (1952) · Zbl 0047.29505
[5]	Kurepa, S., On the Quadratic Functional, Publ. Inst. Math. Acad. Serbe Sci. Beograd, 13, 57-72 (1959) · Zbl 0096.31501
[6]	Rassias, T. M., On the Stability of the Linear Mapping in Banach Spaces, Proc. Amer. Math. Soc., 72, 2, 297-300 (1978) · Zbl 0398.47040
[7]	Ulam, S. M., A Collection of Mathematical Problems (1960), New York: Interscience Publishers Inc., New York · Zbl 0086.24101