Unbounded perturbations of nonlinear discrete periodic problem at resonance (Q1006696)

The authors study the existence of solutions of nonlinear discrete boundary value problems \[ \begin{cases}\Delta^2u(t-1)+\lambda_k u(t)+g(t,u(t))=h(t),\\ u(0)=u(T),\;u(1)=u (T+1),\end{cases}\tag{\(*\)} \] where \(\mathbb T:=[1,\dots,T]\), \(h:\mathbb T\to\mathbb R\), \(\lambda_k\) is the \(k\)-th eigenvalue of the linear problem \[ \begin{cases} \Delta^2u(t-1)+\lambda u(t)=0,\\ u(0)=u(T),\;u(1)=u (T+1),\end{cases}\tag{\(**\)} \] \(g:\mathbb N\times\mathbb R\to\mathbb R\) satisfies some asymptotic nonuniform resonance conditions, and \(g(t,u)u\geq 0\) for \(u\in \mathbb R\). The eigenvalues of the linear problem (\(**\)) are studied in detail. Some examples are considered.