On the existence of m-point boundary value problem at resonance for higher order differential equation

By means of Mawhin's continuation theorem, we study in-point boundary value problem at resonance in the following form: [GRAPHICS] where m greater than or equal to 3, k greater than or equal to 2 are two integers, a(i) epsilon R, xi(i) epsilon (0, 1) (i = 1, 2,..., m - 2) are constants satisfying Sigma(m-1) (i=1) a(i) = 1 and 0 < xi(1) < xi(2) < ... < xi(m-2). A new result on the existence of solutions is obtained. The interesting is that we do not need all the a(i)'s (1 less than or equal to i less than or equal to m - 2) have the same sign, and also the degrees of some variables among x(0), x(1), ...... x(k-1) in the function f (t, x(0), x(1), ..., x(k-1)) are allowable to be greater than 1. Meanwhile, we give some examples to demonstrate our result. (C) 2003 Elsevier Inc. All rights reserved.