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

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 equations

$ x^{(n)}(t)=f(t,x(t),x'(t),\cdots,x^{(n-1)}(t))+e(t),\;\;0<1,\eqno{(\ast)} $

and the following multi-point boundary value conditions

$ \begin{array}{ll} x^{(i)}(0)=0\;\;for\;i=0,1,\cdots,n-3,\\ x^{(n-2)}(0)=\alpha x^{(n-1)}(\xi),\;\;x^{(n-1)}(1)=\beta x^{(n-2)}(\eta),\end{array} \eqno{(\ast\ast)} $

Sufficient conditions for the existence of at least one solution of the BVP$ (\ast) $ and $ (\ast\ast) $ at resonance are established. This paper is directly motivated by Liu and Yu [India J. Pure Appl. Math., 33(4)(2002)475-494] and Qi [Acta Math. Appl. Sinica, 17(2)(2001)271-278].