Abstract. In 1978 P. M. Gruber (Trans. Amer. Math. Soc. 245 (1978), 263--277) imposed the following general problem or Ulam type problem: ``Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this objects by objects, satisfying the property exactly?" The afore-mentioned problem of P. M. Gruber is more general than the following problem imposed by S. M. Ulam in 1940 (Intersci, Publ., Inc., New York 1960): ``Give conditions in order for a linear mapping near an approximately linear mapping to exist". In 1941 D. H. Hyers (Proc. Nat. Acad. Sci., U.S.A. 27 (1941), 411--416) solved a special case of Ulam problem. In 1989 and 1992 we (J. Approx. Th., 57, No. 3 (1989), 268--273; Discuss. Math. 12 (1992), 95--103) solved above Ulam problem. In this paper we introduce the generalized Cauchy-Jensen functional inequality and solve a stability Ulam type problem for this inequality. This problem, according to P. M. Gruber, is of particular interest in probability theory and in the case of functional equations of different types.
Keywords. Ulam problem, Ulam type problem, stability, Cauchy-Jensen, approximately Cauchy-Jensen, Cauchy-Jensen mapping near an approximately Cauchy-Jensen mapping.