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

Yuji Liu, Weigao Ge

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


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.