On the existence and uniqueness of ( N , λ ) (N,λ ) -periodic solutions to a class of Volterra difference equations

In this paper we introduce the class of (N,λ ) -periodic vector-valued sequences and show several notable properties of this new class. This class includes periodic, anti-periodic, Bloch and unbounded sequences. Furthermore, we show the existence and uniqueness of (N, λ ) -periodic solutions to the following class of Volterra difference equations with infinite delay: u(n+1)=α ∑j=-∞ⁿa(n-j)u(j)+f (n,u(n) ), n ∈ Z, α ∈ C, where the kernel a and the nonlinear term f satisfy suitable conditions.