Asymptotic behavior of solutions of second-order difference equations of Volterra type

Abstract

In this paper we investigate the Volterra difference equation of the form $triangle(r_ntriangle x_n)=b_n+sum_{k+1}^nK(n,k)f(x_k)$ 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 $triangle(r_ntriangle y_n)=b_n$ and s is nonpositive real number. We also obtain sufficientconditions for the existence of asymptotically periodic solutions.