In this paper we investigate the Volterra difference equation of the form $ \D(r_n\D x_n)=b_n+\sum_{k=1}^{n}K(n,k)f(x_k). $ We establish sufficient conditions for the existence of a solution $x$ of the above equation with the property $ x_n=y_n+\o(n^s), $ where $y$ is a given solution of the equation $\D(r_n\D y_n)=b_n$ and $s$ is nonpositive real number. We also obtain sufficient conditions for the existence of asymptotically periodic solutions.