On the Ulam problem for Euler quadratic mappings

John Michael Rassias

In 1940 and in 1968 S. M. Ulam proposed the general problem: "When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?". In 1941 D. H. Hyers solved this stability problem for linear mappings. In 1951 D. G. Bourgin was the second author to treat the same problem for additive mappings. According to P. M. Gruber (1978) this kind of stability problems are of particular interest in probability theory and in the case of functional equations of different types. In 1982-2002 we solved the above Ulam problem for linear and non-linear mappings and established analogous stability problems even on restricted domains. Besides, we applied some of our recent results to the asymptotic behavior of functional equations of different types. In this paper we investigate the Euler quadratic mappings $Q\colon X\to Y$, satisfying the functional equation \begin{multline*} Q(x_0-x_1)+Q(x_1-x_2)+Q(x_2-x_3)+Q(x_3-x_0) =Q(x_0-x_2)+Q(x_1-x_3)+Q(x_0-x_1+x_2-x_3) \end{multline*} and then solve the corresponding Ulam stability problem.