Asymptotically periodic solutions of Volterra difference equations (Q1411127)

For the difference equations \[ x_{n+1}= a(n)- \sum^n_{k=0} D(n,k,x_k),\qquad n\in\mathbb{Z}^+, \] and \[ x_{n+1}= p(n)- \sum^n_{k=-\infty} P(n,k,x_k),\qquad n\in\mathbb{Z}, \] under certain conditions the existence of asymptotically periodic solutions is proved. Moreover, several relations between solutions of the two equations are discussed, as well as the existence of periodic solutions.