Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 24, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF SOLUTIONS FOR SECOND-ORDER IMPULSIVE BOUNDARY-VALUE PROBLEMS ABDELKADER BOUCHERIF, ALI S. AL-QAHTANI, BILAL CHANANE Abstract. In this article we discuss the existence of solutions of second-order boundary-value problems subjected to impulsive effects. Our approach is based on fixed point theorems. 1. Introduction Differential equations involving impulse effects arise naturally in the description of phenomena that are subjected to sudden changes in their states, such as popula- tion dynamics, biological systems, optimal control, chemotherapeutic treatment in medicine, mechanical systems with impact, financial systems. For typical examples see [9, 11]. For a general theory on impulsive differential equations the interested reader can consult the monographs [2, 7, 14], and the papers [1, 5, 6, 8, 10, 12, 13, 15] and the references therein. Our objective is to provide sufficient conditions on the data in order to ensure the existence of at least one solution of the problem (p(t)x  (t))  + q(t)x(t)= F (t, x(t),x  (t)), t = t k ,t ∈ [0, 1], Δx(t k )= U k (x(t k ),x  (t k )), Δx  (t k )= V k (x(t k ),x  (t k )), k =1, 2,...,m, x(0) = x(1) = 0, (1.1) where x ∈ R is the state variable; F : R + × R 2 → R is a piecewise continuous function; U k and V k represent the jump discontinuities of x and x  , respectively, at t = t k ∈ (0, 1), called impulse moments, with 0 <t 1 <t 2 < ··· <t m < 1. 2. Preliminaries In this section we introduce some definitions and notations that will be used in the remainder of the paper. Let J denote the real interval [0, 1]. Let J  = J \{t 1 , t 2 ,...,t m }. PC(J ) denotes the space of all functions x : J → R continuous on J  , and for i =1, 2,...,m, x(t + i ) = lim →0+ x(t i + ) and x(t - i ) = lim →0 x(t i - ) exist. We shall write x(t - i )= x(t i ). This is a Banach space when equipped with the sup-norm; i.e., 2000 Mathematics Subject Classification. 34B37, 34B15, 47N20. Key words and phrases. Second order boundary value problems; impulse effects; fixed point theorem. c 2012 Texas State University - San Marcos. Submitted September 12, 2011. Published February 7, 2012. 1