Existence of periodic solutions for a 2nth-order nonlinear difference equation

The authors consider the 2nth-order difference equationΔn(rt−nΔnxt−n)+f(t,xt)=0,n∈Z(3),t∈Z, where f:Z×R→R is a continuous function in the second variable, f(t+T,z)=f(t,z) for all (t,z)∈Z×R, rt+T=rt for all t∈Z, and T a given positive integer. By the Linking Theorem, some new criteria are obtained for the existence and multiplicity of periodic solutions of the above equation.