Multiple positive solutions for a nonlinear 2n-th order m-point boundary value problems

In this paper, we consider the existence of multiple positive solutions for the 2n-th order $m$-point boundary value problems: $$\left\{\begin{array}{ll} x^{(2n)}(t)=f(t,x(t),x^{''}(t),\cdots ,x^{(2(n-1))}(t)), 0\leq t\leq 1,\\ x^{(2i+1)}(0)=\sum\limits_{j=1}^{m-2}\alpha_{ij}x^{(2i+1)}(\xi_j),\quad x^{(2i)}(1)=\sum\limits_{j=1}^{m-2}\beta_{ij}x^{(2i)}(\xi_j), 0\leq i\leq n-1,\\ \end{array}\right.$$ where $\alpha_{ij}, \beta_{ij} \ (0\leq i\leq n-1,1\leq j\leq m-2) \in [0,\infty)$, $\sum\limits_{j=1}^{m-2}\alpha_{ij},\sum\limits_{j=1}^{m-2}\beta_{ij}\in (0,1)$, $0<\xi_1<\xi_2<\ldots<\xi_{m-2}<1$. Using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem