Solution of a Cauchy-Jensen stability Ulam type problem. (English) Zbl 1090.39014

Summary: In 1978 P. M. Gruber [Trans. Am. Math. Soc. 245, 263–277 (1978; Zbl 0393.41020)] 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 [A collection of mathematical problems (Interscience Tracts in Pure and Applied Mathematics 8, New York) (1960; Zbl 0086.24101)]: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.” In 1941 D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; Zbl 0061.26403)] solved a special case of Ulam problem. In 1989 and 1992 we 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.


39B62 Functional inequalities, including subadditivity, convexity, etc.
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