Positive solution for a class of nonlinear fourth-order boundary value problem

In this paper, we are concerned with the existence of positive solutions for boundary value problems of nonlinear fourth-order differential equations

$$\begin{array}{rcl}& & u^{(4)}+c(x)u=\lambda a(x)f(u),x\in (0,1),\\ & & u(0)=u(1)=u^{″}(0)=u^{″}(1)=0,\end{array}$$

where $a(x)$ may change signs. The proof of main results is based on Leray-Schauder's fixed point theorem and the properties of Green's function of the fourth-order differential operator $L_{c}u=u^{(4)}+c(x)u$.