Bifurcation techniques and positive solutions of discrete Lidstone boundary value problems

Let n ⩾ 1, T ⩾ n + 5 be two positive integers and Tn={n,n+1,…,T}Tn={n,n+1,…,T}. We consider the existence of solutions of the discrete Lidstone boundary value problem.(-1)nΔ2nu(t-n)=f(t,u(t),Δ2u(t-1),…,Δ2n-2u(t-n+1)),t∈Tn,Δ2iu(n-i-1)=Δ2iu(T-i+1)=0,i=0,1,…,n-1,where f:Tn×Rn→Rf:Tn×Rn→R is continuous.