SOLUTION OF A QUADRATIC STABILITY ULAM TYPE PROBLEM

mapping near an approximately linear mapping to exist". According to P. M. Gru-ber (Trans. Amer. Math. Soc. 245 (1978), 263{277) the afore-mentioned problem of S. M. Ulam belongs to the following general problem or Ulam type problem: \Sup-pose a mathematical object satises a certain property approximately. Is it then possible to approximate this objects by objects, satisfying the property exactly?" In 1941 D. H. Hyers (Proc. Nat. Acad. Sci. 27 (1941), 411{416) established the stability Ulam problem with Cauchy inequality involving a non-negative constant. Then in 1989 we (J. Approx. Theory, 57 (1989), 268{273) solved Ulam problem with Cauchy functional inequality, involving a product of powers of norms. Finally we (Discuss. Math. 12 (1992), 95{103) established the general version of this sta-bility problem. In this paper we solve a stability Ulam type problem for a general quadratic functional inequality. Moreover, we introduce an approximate eveness on approximately quadratic mappings of this problem. These problems, according to P. M. Gruber (1978), are of particular interest in probability theory and in the case of functional equations of dierent types. Today there are applications in actuar-ial and nancial mathematics, sociology and psychology, as well as in algebra and geometry. De nition 1. Let X be a linear space and also let Y be a real linear space. Then a mapping Q2: X! Y is called quadratic, if the functional equatio