Abstract :
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
