Application of Lyapunov’s Direct Method to the Existence of Integral Manifolds of Impulsive Differential Equations

The present paper investigates the existence of integral manifolds for impulsive differential equations with variable perturbations. By means of piecewise continuous functions which are generalizations of the classical Lyapunov’s functions, sufficient conditions for the existence of integral manifolds of such equations are found. 1. Preliminary notes and definitions. In recent years, the impulsive differential equations have been an object of numerous investigations (Bainov and Simeonov, 1989; Dishliev and Bainov, 1989; Lakshmikantham, Bainov and Simeonov, 1989; Kulev and Bainov 1990; Simeonov and Bainov, 1991, 1993) In the present paper, some problems related to the existence of an integral manifold are considered. The main results are obtained by means of piecewise continuous functions which are analogous to the classical Lyapunov’s functions. Let R be the n-dimensional Euclidean space with norm ‖·‖ and scalar product 〈·, ·〉, and let I = [0,∞). We denote by PC(J,R), where J ⊂ I, k = 1, 2, . . ., the space of all piecewise continuous functions x : J → R such that: 1991 Mathematics Subject Classification: 34A37