Unbounded perturbations of nonlinear discrete periodic problem at resonance

In this paper, we study the existence of solutions of nonlinear discrete boundary value problems {Δ2u(t−1)+λku(t)+g(t,u(t))=h(t),u(0)=u(T),u(1)=u(T+1), where T≔{1,…,T}T≔{1,…,T}, h:T→Rh:T→R, λkλk is the kk-th eigenvalue of the linear problem {Δ2u(t−1)+λu(t)=0,u(0)=u(T),u(1)=u(T+1),g:N×R→Rg:N×R→R satisfies some ‘asymptotic nonuniform’ resonance conditions, and g(t,u)u≥0g(t,u)u≥0 for u∈Ru∈R.