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Ulam problem; Ulam type problem; stability; quadratic; approximate eveness; approximately quadratic; quadratic mapping near an approximately quadratic mapping
In 1940 S. M. Ulam (Intersci. Publ., Inc., New York 1960) imposed at the University of Wisconsin the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist”. According to P. M. Gruber (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: “Suppose a mathematical object satisfies 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 stability 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 different types. Today there are applications in actuarial and financial mathematics, sociology and psychology, as well as in algebra and geometry.
[1] Gruber, P. M.: Stability of Isometries. Trans. Amer. Math. Soc. 245 (1978), 263–277. MR 0511409 | Zbl 0393.41020
[2] Hyers, D. H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. 27 (1941), 411–416. MR 0004076 | Zbl 0844.39001
[3] Rassias, J. M.: Solution of a problem of Ulam. J. Approx. Theory 57 (1989), 268–273. MR 0999861 | Zbl 0672.41027
[4] Rassias, J. M.: Solution of a stability problem of Ulam. Discuss. Math. 12 (1992), 95–103. MR 1221875 | Zbl 0878.46032
[5] Ulam, S. M.: A collection of mathematical problems. Intersci. Publ., Inc., New York, 1960. MR 0120127 | Zbl 0086.24101
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