Stability and boundedness in functional differential equations

Abstract

Among the methods in Liapunov stability theory for ordinary and functional differential equations, the comparison method is quite useful. In this method, Liapunov functions (or functionals) play important roles. Corresponding to Liapunov functions for ordinary differential equations, Liapunov functionals have been used naturally for functional differential equations. However, sometimes it is not easy to construct a suitable Liapunov functional for a given functional differential equation. To avoid such a difficulty and obtain stability results for functional differential equations, many efforts have been made by using Liapunov functions instead of Liapunov functionals [2, 66131. The purpose of this paper is to employ the theory of Liapunov-Razumikhin type to study the stability and boundedness of solutions of functional differential equations. The comparison method used in this paper is similar to the one used in [3-51 to show the existence of periodic solutions of functional differential equations. In Section 5, we present some applications of the results obtained in Sections 3 and 4 to integrodifferential equations of Volterra type.