Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 120, pp. 1-19.

Title: Solvability of a (p, n-p)-type multi-point boundary-value problem
       for higher-order differential equations

Authors: Yuji Liu (Beijing Inst. of Technology, China)
 Weigao Ge (Beijing Inst. of Technology, China)

Abstract: 
 In this article, we study the differential equation
 $$
 (-1)^{n-p} x^{(n)}(t)=f(t,x(t),x'(t),\dots,x^{(n-1)}(t)),\;\;0<t<1,
 $$
 subject to the multi-point boundary conditions
 $$\displaylines{
 x^{(i)}(0)=0 \quad \hbox{for }i=0,1,\dots,p-1,\cr
 x^{(i)}(1)=0 \quad \hbox{for }i=p+1,\dots,n-1,\cr
 \sum_{i=1}^m\alpha_ix^{(p)}(\xi_i)=0,
 }$$
 where $1\le p\le n-1$. We establish sufficient conditions for the
 existence of at least one solution at resonance and another at
 non-resonance. The emphasis in this paper is that $f$ depends on
 all higher-order derivatives. Examples are given to illustrate the
 main results of this article.

Submitted August 19, 2003. Published December 1, 2003.
Math Subject Classifications: 34K20,  92D25
Key Words: Solvability; resonance; non-resonance; 
multi-point boundary-value problem; higher order differential equation.