Nonhomogeneous boundary value problem for first-order impulsive differential equations with delay

K e y w o r d s N o n h o m o g e n e o u s boundary value problem, Monotone iterative technique, Lower solution, Upper solution, Impulsive differential equations with delay. 1. I N T R O D U C T I O N We are concerned w i t h the fol lowing n o n h o m o g e n e o u s b o u n d a r y va lue p r o b l e m for a f i r s t -order impuls ive dif ferent ia l equa t i on w i t h delay in R, x' (t) = f (t, x ( t ) , z t ) , Ax (tk) = Ik (~ ( tk) ) , x (0) = • (0) , x (o) ~ (T) = ~ e n , t E J', k = 1 , 2 , . . . , m , for a g iven 0 E [--T, 0), (1.1) where f : J × R × D --* R, D -L I ( [ T , 0 ] ,R) , Ik 6 C ( R , R ) , A x ( t k ) r ep resen t s t he j u m p of x ( t ) at t = tk, i.e., A x ( t k ) = x ( t +) x(t~-), for all k = 1 , 2 , . . . , m , 0 < t l < t2 < . . < t,~ < T, 5 = m a x ( t k tk -1 ; k -1 , 2 , . . . , m + 1} here to -O, tm+Z = T; T > O, J =[0, T], J ' = J \ { t l , t 2 , . . . , t m } ; for every t e J , xt e D is def ined by x t ( s ) = x ( t + s), ' r < s < O. This work is supported by the National Sciences Foundation of China (No. 10471040) and the Sciences Foundation of Shanxi (No. 2005Z010) and the Major Subject Foundation of Shanxi (20055024). The authors thank the referee for his valuable suggestion. 0898-1221/06/$ see front matter (~) 2006 Elsevier Ltd. All rights reserved. Typeset by A~4S-TF_~ doi:10.1016/j.camwa.2005.11.028