Nontrivial Periodic Solutions to Some Semilinear Sixth-Order Difference Equations

Abstract

We establish some new criteria to guarantee nonexistence, existence, and multiplicity of nontrivial periodic solutions of some semilinear sixth-order difference equations by using minmax method, $Z_{\mathrm{2}}$ index theory, and variational technique. Our results only make some assumptions on the period $T$, which are very easy to verify and rather relaxed.