Existence of Periodic Solutions to Nonlinear Difference Equations with Delay

This paper deals with the existence and multiplicity of nontrivial periodic solutions to the first order nonlinear difference equation with delay of the form△x(n)=-f(x(n-T)) By applying the critical point theory,several sufficient conditions are obtained for the periodic solutions to the equation above under the assumption that f satisfies certain growth conditions both at zero and at infinity.Some examples are given to illustrate the feasibility of the main results.