Let \(T\in {\mathbb N}\) be an integer with \(T>2\), and \({\mathbb T}:=\{1,2,\dots,T\}\). The authors study the existence of solutions of nonlinear discrete problems \(\Delta ^{2}u(t-1)+\lambda _{k}a(t)u(t)+g(t,u(t))=h(t),\) \(t\in {\mathbb T},\) \(u(0)=u(T),\) \(u(1)=u(T+1),\) where \(a,\) \(h:{\mathbb T}\rightarrow {\mathbb R}\) with \(a>0,\) \(\lambda _{k}\) is the \(k\)-th eigenvalue of the corresponding linear eigenvalue problem. Some examples are considered.