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 maps a normed space into a Banach space and satisfies the inequality
with some and find a (possibly unique) quadratic mapping lying “not far” from .
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 , or , and .
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 the quadratic functional equation is not stable in the considered sense.