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John M. Rassias

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*Solution of a Cauchy-Jensen stability Ulam type problem*

**Address.** National and Capodistrian University of Athens, Pedagogical
Department E.E.,

Section of Mathematics and Informatics, 4, Agamemnonos Str., Aghia
Paraskevi,

Attikis 15342, GREECE

**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.

**AMSclassification.** 39B

**Keywords.** Ulam problem, Ulam type problem, stability, Cauchy-Jensen,
approximately Cauchy-Jensen, Cauchy-Jensen mapping near an approximately
Cauchy-Jensen mapping.