Asymptotic behavior of Jensen and Jensen type functional equations

In 1941 D.H. Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 D.G. Bourgin was the second author to treat the Ulam problem for additive mappings. In 1982-2004 we established the Hyers-Ulam stability for the Ulam problem of linear and nonlinear mappings. In 1983 F. Skof was the first author to solve the Ulam problem for additive mappings on a restricted domain. In 1998 S.-M. Jung and in 2002-2004 the authors of this paper investigated the Hyers-Ulam stability of additive and quadratic mappings on restricted domains. In this paper we improve our bounds and thus our results obtained,in 2003 for Jensen and Jensen type mappings and establish new theorems about the Ulam stability of additive mappings of the first (and the second) form on restricted domains. Finally, we apply our recent results to the asymptotic behavior of functional equations of these types. 22 John Rassias and Matina Rassia