Existence of solution for a class of nth-order multi-point boundary value problem

Abstract

In this paper, we are concerned with the following nth-order ordinary differential equation

$$x^{(n)}(t)+f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))=0,\quad t\in (0,1),$$

with the nonlinear boundary conditions

$$\begin{array}{l}x^{(i)}(0)=0,\quad i=0,1,\ldots,n-3,\\[3pt]g(x^{(n-2)}(0),x^{(n-1)}(0),x(\xi_1),\ldots,x(\xi_{m-2}))=A,\\[3pt]h(x^{(n-2)}(1),x^{(n-1)}(1),x(\eta_1),\ldots,x(\eta_{l-2}))=B,\end{array}$$

here A,B∈R, f:[0,1]×R ⁿ→R is continuous, g:[0,1]×R ^m→R is continuous, h:[0,1]×R ^l→R is continuous, ξ _i ∈(0,1), i=1,…,m−2, and η _j ∈(0,1), j=1,…,l−2. The existence result is given by using a priori estimate, Nagumo condition, the method of upper and lower solutions and Leray-Schauder degree. We also give an example to demonstrate our result.