Matina John Rassias

Khasminskii-Type Theorems for Stochastic Differential Delay Equations

Abstract The classical Khasminskii theorem (see [6]) on the nonexplosion solutions of stochastic differential equations (SDEs) is very important since it gives a powerful test for SDEs to have nonexplosion solutions without the linear growth condition. Recently, Mao [13] established a Khasminskii-type test for stochastic differential delay equations (SDDEs). However, the Mao test can not still be applied to many important SDDEs, e.g., the stochastic delay power logistic model in population dynamics. The main aim of this paper is to establish an even more general Khasminskii-type test for SDDEs that covers a wide class of highly nonlinear SDDEs. As an application, we discuss a stochastic delay Lotka-Volterra model of the food chain to which none of the existing results but our new Khasminskii-type test can be applied.

On the Ulam stability of Jensen and Jensen type mappings on restricted domains

In 1941 Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 Bourgin was the second author to treat this problem for additive mappings. In 1982–1998 Rassias established the Hyers–Ulam stability of linear and nonlinear mappings. In 1983 Skof was the first author to solve the same problem on a restricted domain. In 1998 Jung investigated the Hyers–Ulam stability of more general mappings on restricted domains. In this paper we introduce additive mappings of two forms: of “Jensen” and “Jensen type,” and achieve the Ulam stability of these mappings on restricted domains. Finally, we apply our results to the asymptotic behavior of the functional equations of these types.

A Fixed Point Approach to the Stability of Quintic and Sextic Functional Equations in Quasi--Normed Spaces

We achieve the general solution of the quintic functional equation and the sextic functional equation . Moreover, we prove the stability of the quintic and sextic functional equations in quasi--normed spaces via fixed point method.