Periodic solutions of nonlinear functional difference equations at nonresonance case

Sufficient conditions for the existence of at least one T-periodic solution of nonlinear functional difference equationΔx(n)+a(n)x(n)=f(n,u(n)), is established when ∏j=0T−1(1−a(j))≠1. Hereu(n)=(x(n),x(n−τ1(n)),…,x(n−τm(n))), {a(n):n∈Z} and {τi(n):n∈Z}, i=1,…,m, are T-periodic sequences with T⩾1, f(n,u) is continuous about u for each n∈Z and T-periodic about n for each u∈Rm+1. We allow f to be at most linear, superlinear or sublinear in obtained results.