I. Introduction
In recent years, because of the wide mathematical and physical background, the existence of positive solution for sub-linear boundary value problems received wide attention. Recently, for the existence of positive solutions of second-order sub-linear boundary problems, some authors have obtained the existence results (see [4]-[10]). However, with the author's knowledge, there are few papers that have studied the positive solutions for the four-point boundary value problem with singularities. Inspired by paper[7,9-11], in this paper, we research the existence of positive solutions of the following singular four-point boundary value problems:\begin{align*} &\ -u^{\prime\prime}(t)=f(t,u(t)),\ t\in(0,1),\tag{1}\\ &u(0)=au(\xi), u(1)= bu(\eta),\tag{2} \end{align*} where satisfying the following assumptions:\begin{align*} &\ (\mathrm{H})\\ &f\in C([0,1]\times(0,+\infty),[0,+\infty)),\\ &f(t,1) > 0,t\in(\mathrm{O},1), \end{align*} and there exists constants such that for , we have\begin{equation*} c^{\mu}f(t,u)\leq f(t,cu)\leq c^{\lambda}f(t,u),0 < c\leq 1, \tag{3} \end{equation*} where satisfy that ‘