Monotone iterative for fourth-order p-Laplacian boundary value problems with impulsive effects q Guoliang Shi * , Xianrui Meng Department of Mathematics, Tianjin University, Tianjin 300072, PR China Abstract Sufficient conditions are given for the existence of a solution of fourth-order p-Laplacian boundary value problems with impulsive effects. The conditions assume the nonlinear function in equation is monotone, and the existence of an upper– lower solution pair. The properties of p-Laplacian operator are employed. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Upper and lower solutions; The fourth-order; p-Laplacian equations; Impulsive 1. Introduction In this paper, we are concerned with the existence behavior of solutions for the following fourth-order p-Laplacian Lidstone BVPs with the impulsive effects, ðu p ðx 00 ðtÞÞÞ 00 ¼ f ðt; xðtÞ; x 00 ðtÞÞ; t 2ðt i ; t iþ1 Þ; i ¼ 0; 1; ... ; k; xð0Þ¼ xð1Þ¼ x 00 ð0Þ¼ x 00 ð1Þ¼ 0; Dxðt i Þ¼ a 1 ðiÞ; Dx 0 ðt i Þ¼ a 2 ði; xðt i Þ; x 00 ðt i ÞÞ; Duðx 00 ðt i ÞÞ ¼ a 3 ðiÞ; Duððx 00 ðt i ÞÞÞ 0 ¼ a 4 ði; xðt i Þ; x 00 ðt i ÞÞ; 8 > > > > > > > > < > > > > > > > > : ð1:1Þ where 0= t 0 < t 1  < t k < t k+1 = 1, u p (u)= juj p2 u, 1< p < 1, and f : [0, 1] · R 2 ! R is continuous, a l : R 2 ! R, l = 2, 4 is continuous, a 1 (i) and a 3 (i) are constants, 1 6 i 6 k. x(t + ) = lim s!t+0 x(s) and x(t  ) = lim s!t0 x(s) exist, Dx(t)= x(t + )  x(t  ) and xðt i Þ¼ xðt  i Þ, i = 1, 2, ... , k. It is easy to be verified that the inverse function of u p is u q , where q ¼ p p1 . Moreover, u p (u) and u q (u) are increasing functions with respect 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.02.024 q This research was supported by Nankai Univ. and Tianjin Univ. Liuhui Center of Applied Mathematics. * Corresponding author. E-mail address: glshi@tju.edu.cn (G. Shi). Applied Mathematics and Computation 181 (2006) 1243–1248 www.elsevier.com/locate/amc