##
Solutions of a multi-point boundary value problem for
higher-order differential equations at
resonance (II)

##
*Yuji Liu, Weigao Ge*

**Address.**

Department of Applied Mathematics,
Hunan Institute of Science and Technology,
Hunan, 414000, P.R.China

Department of Mathematics,
Beijing Institute of Technology,
Beijing, 100081, P.R.China

**E-mail. **liuyuji888@sohu.com

**Abstract.**

In this paper, we are concerned with the existence of solutions of
the following multi-point boundary value problem consisting of the
higher-order differential equation
$$
x^{(n)}(t)=f(t,x(t),x'(t),\dots,x^{(n-1)}(t))+e(t)\,,\quad 0< t<1\,,\leqno{(\ast)}
$$
and the following multi-point boundary value conditions
\begin{align*}
x^{(i)}(0)&=0\quad \mbox{for}\quad i=0,1,\dots,n-3\,,\\
x^{(n-1)}(0)&=\alpha
x^{(n-1)}(\xi)\,,\quad
x^{(n-2)}(1)=\sum_{i=1}^m\beta_ix^{(n-2)}(\eta_i)\,.\tag{**}
\end{align*}
Sufficient conditions for the existence of at least one solution
of the BVP $(\ast)$ and $(\ast\ast)$ at resonance are established.
The results obtained generalize and complement those in [13, 14].
This paper is directly motivated by Liu and Yu [J. Pure Appl.
Math. 33 (4)(2002), 475--494 and Appl. Math. Comput.
136 (2003), 353--377].

**AMSclassification. **34B15.

**Keywords. ** Solution, resonance, multi-point boundary value
problem, higher order differential equation.