ftp ejde.math.txstate.edu (login: ftp) IMPULSIVE DIFFERENTIAL INCLUSIONS WITH CONSTRAINS

Abstract. In the paper, we study weak invariance of differential inclusions with non-fixed time impulses under compactness type assumptions. When the right-hand side is one sided Lipschitz an extension of the well known relaxation theorem is proved. In this case also necessary and sufficient condition for strong invariance of upper semi continuous systems are obtained. Some properties of the solution set of impulsive system (without constrains) in appropriate topology are investigated. 1. Preliminaries This paper is concerned with the impulsive differential inclusion ẋ(t) ∈ F (t, x(t)), x(0) = x0 ∈ D, a.e. t ∈ I = [0, 1], t 6 = τi(x), (1.1) ∆x|t=τi(x) = Si(x), i = 1,..., p, x(t) ∈ D, (1.2) Here D is a closed subset of a Banach space E and F: I ×D → E is multifunction with nonempty compact values. Every absolutely continuous on (τi, τi+1) function for i = 0, 1,..., p, p + 1 (τ0 = 0 and τp+1 = 1) with (possible) jumps ∆x|t=τi(x) = Si(x(τi(x) − 0)) called impulses, i.e. x(τi(x) + 0) = x(τi(x) − 0) + Si(x(τi(x) − 0), is said to be a solution of (1.1)–(1.2). Further we assume that x+ Si(x) ∈ D for every x ∈ D. Differential inclusions without impulses have been studied extensively; see [2, 3