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 m-point boundary value problem at resonance in the following form: x(k)(t ) = f (t,x(t),x (t ), . . . , x(k−1)(t ))+ e(t ), t ∈ (0, 1), x (0) = 0, x (0) = 0, . . . , x(k−1)(0) = 0, x(1) = m−2 i=1 aix(ξi ), where m 3, k 2 are two integers, ai ∈ R, ξi ∈ (0, 1) (i = 1, 2, . . . , m−2) are constants satisfying m−1 i=1 ai = 1 and 0 <ξ1 <ξ2 < ··· < ξm−2. A new result on the existence of solutions is obtained. The interesting is that we do not need all the ai ’s (1 i m − 2) have the same sign, and also the degrees of some variables among x0,x1, . . . , xk−1 in the function f (t,x0,x1, . . . , xk−1) are allowable to be greater than 1. Meanwhile, we give some examples to demonstrate our result.  2003 Elsevier Inc. All rights reserved