Czerwik, St. On the stability of the quadratic mapping in normed spaces. (English) Zbl 0779.39003 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) Cited in 4 ReviewsCited in 292 Documents 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 PDF BibTeX XML Cite \textit{St. Czerwik}, Abh. Math. Semin. Univ. Hamb. 62, 59--64 (1992; Zbl 0779.39003) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.