## Abstract

By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation

$Δ2xn-1+f(n,xn)=0,$

where fC(R × Rm, Rm), f(t+M,z)+f(t,z) for any (t, z)∈R × Rm and M is a positive integer. This is probably the first time critical point theory has been applied to deal with the existence of periodic solutions of difference systems.