Periodic solutions for difference systems with delay

In this work we consider the existence of periodic solutions for the difference system of the form x(n+1)=A(n)x(n)+F(n,x(n),x(σ(n)))+C(n)x(n+1)=A(n)x(n)+F(n,x(n),x(σ(n)))+C(n). First we establish some stability criteria for it. Then, by making use of Horn’s fixed point theorem, we obtain a criterion to guarantee the existence.