Solvability of a nonlinear fourth-order discrete problem at resonance

Let T   be an integer with T⩾5T⩾5 and let T2={2,3,…,T}T2={2,3,…,T}. We consider the nonlinear discrete boundary value problemΔ4u(t-2)=λ1u(t)+f(t,u(t))+τφ(t)+h¯(t),t∈T2,u(1)=u(T+1)=Δ2u(0)=Δ2u(T)=0,where λ1λ1 is the first eigenvalue of the associated linear eigenvalue problem, φ(·)φ(·) is the corresponding eigenfunction; f:T2×R→Rf:T2×R→R is continuous and|f(t,s)|⩽A|s|α+B,t∈T2,s∈Rfor some 0⩽α<10⩽α<1 and A,B∈[0,∞);h¯:T2→R with ∑s=2Th¯(t)φ(t)=0. We show the existence of solutions of the above problem. Our approaches are based on the Krein–Rutman theorem, connectivity properties of solution sets of parameterized families of compact vector fields.