In this paper, we study the following periodic boundary value problem of fourth-order ordinary differential equation

\ \begin{equation*}

\ \left\{

\ \ \begin{aligned}

\ \ \&u^{(4)}(t)+\alpha u^{\prime\prime}(t)-\rho^{4}u(t)+\lambda f(t,u(t))=0,~t\in{[0,2\pi]},\\

\ \ \&u^{(i)}(0)=u^{(i)}(2\pi),~i=0,1,2,3,\\

\ \ \end{aligned}

\ \ \right.

\ \end{equation*}

where $\alpha$ and $\rho$ are constants satisfying $\rho\neq0$ and $4\alpha+16\rho^{4}\<1$, and $\lambda\>0$ is a parameter. By imposing some conditions on the nonlinear term $f$, we obtain the existence and multiplicity of positive solutions to the above problem for suitable $\lambda$. The main tool used is Guo-Krasnoselskii fixed point theorem.