Nontrivial solutions for nonlinear discrete boundary value problems of the fourth order

We study the existence of multiple nontrivial solutions for nonlinear fourth order discrete boundary value problems. first establish criteria at least two problems and obtain conditions to guarantee that are sign-changing. Under some appropriate assumptions, we further prove have three solutions, which positive, negative, sign-changing, respectively. include examples illustrate applicability our results. Our theorems proved by employing variational approaches, combined with classic mountain pass lemma a result from theory invariant sets descending flow.