State-dependent impulsive delay-differential equations

K e y w o r d s D e l a y , Differential equation, Impulses, Fixed point. 1. I N T R O D U C T I O N The object of this paper is to present existence and uniqueness results about the solution of a system of delay-differential equations with infinitely many state-dependent impulses. This type of problem is characterized by jumps in the solution of the system. The system is (s) = / ( t , x , ) + ( V ) ) jeN x(0) = ¢0, X(O) = •1(0), t E [0, T], 0 e I ( h , 0), where h, 0 < h < +co is the length of the memory of the system, I ( h , 0) = I -h , 0] n R, and for all t > 0, the function xt is defined on I ( h , 0) by xt(O) = x(t+0) ; x : I ( h , T) * R n is a vector This work has been supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Grant). 0893-9659/02/$ see front matter © 2002 Elsevier Science Ltd. All rights reserved. Typeset by ~4.£4S-TEX PII: S0893-9659(01)00140-9 334 F. DUBEAU AND J. KARRAKCHOU function, (¢o, ¢1) • R,~ x LP(-h, 0; R '~) is the initial condition, f : [0, T] × K ( h , 0; R n) is a given map where { C ( h , 0 ; R ~ ) , if h < +co, K(-h'O;R'~)= Co( -h , 0; R'~), if h = +c~, , R" 5(.) is the Dirac delta function at 0, and for each j • N, s t : IR n , R n is a given map, Tj real > 0, ~ real > h , and ¢j ~ Tj. The case of a system of ordinary differential equations was studied recently by Dubeau et al. [1]. The principal result of this paper is a complement to the existing literature (see, for example, [2-5]) and generalizes the result given in [1]. 2. P R E L I M I N A R Y R E S U L T S A solution of system (S) is a function of bounded variation which can be written in the form