Summary: We consider the fourth order periodic boundary value problem \[ \begin{aligned}&\Delta^4u(t-2)-\Delta (p(t-1)\Delta u(t-1))+q(t)u(t)=f(t,u(t)),\quad t\in [1,N]_\mathbb Z, \\ &\Delta^iu(-1)= \Delta^iu(N-1),\quad i=0,1,2,3,\end{aligned} \] where \(N\geq 1\) is an integer, \(p\in C([0,N]_\mathbb Z,\mathbb R)\), \(q\in C([1,N]_\mathbb Z,\mathbb R)\), and \(f\in C([1,N]_\mathbb Z\times\mathbb R,\mathbb R)\). We obtain sufficient conditions for the existence of one and two solutions of the problem. The analysis is based mainly on the variational method and critical point theory.