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TWO-POINT BOUNDARY VALUE PROBLEMS FOR HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH STRONG

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(i−1) + q(t) (1.1) with the conjugate boundary conditions u(i−1) (a) = 0 (i =. | 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 u n pi t u i-1 q t 1.1 with the conjugate boundary conditions u i-1 a 0 i 1 . m u j-1 b 0 j 1 . n - m 1.2 or the right-focal boundary conditions u i-1 a 0 i 1 . m u j-1 b 0 j m 1 . n . 1.3 Here n 2 m is the integer part of n 2 - a b TO pi e Lloc a b i 1 . n q e Lloc a b and by u i-1 a by u j-1 b is understood the right the left limit of the function u i-1 of the function u j-1 at the point a at the point b . Problems 1.1 1.2 and 1.1 1.3 are said to be singular if some or all coefficients of 1.1 are non-integrable on a b having singularities at the ends of this segment. Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID 83910 Pages 1-32 DOI 10.1155 BVP 2006 83910 2 Linear BVPs with strong singularities The previous results on the unique solvability of the singular problems 1.1 1.2 and 1.1 1.3 deal respectively with the cases where b t - a n-1 b - t 2m-1 - 1 n-mp1 t dt a b t - a n-i b - t 2m-i pi t dt TO i 2 . m a b t - a n-m-1 2 b - t m-1 2 I q t I dt a b t - a n-1 -1 n-mp1 t dt a b t - a n-i pi t dt TO i 2 . m a b t - a n-m-1 2 q t dt a 1.4 1.5 see 1 2 4 3 5 6 9-18 and the references therein . The aim of the present paper is to investigate problem 1.1 1.2 problem 1.1 1.3 in the case where the functions pi i 1 . n and q have strong singularities at the points a and b at the point a and do not satisfy .