Báo Cáo Báo cáo khoa học Two-point boundary value problems for higher-order linear differential equations wi

TWO-POINT BOUNDARY VALUE PROBLEMS FOR HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH STRONG SINGULARITIES R. P. AGARWAL AND I. KIGURADZE Received 4 April 2004; Revised 11 December 2004; Accepted 14 December 2004 For strongly singular higher-order linear differential equations together with two-point conjugate and right-focal boundary conditions, we provide easily verifiable best possible conditions which guarantee the existence of a unique solution. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Statement of the main results 1.1. Statement of the problems and the basic notation. Consider the differential equation m u(n) = i=1 pi (t)u(iư1) + q(t) (1.1) with the conjugate boundary conditions u(iư1) (a) = 0 (i = 1, .,m), u( j ư1) (b) = 0 ( j = 1, .,n ư m) or the right-focal boundary conditions u(iư1) (a) = 0 u( j ư1) (b) = 0 (i = 1, .,m), ( j = m + 1, .,n). (1.3) (1.2) Here n ≥ 2, m is the integer part of n/2, ư∞