Asymptotic properties of solutions to discrete Volterra type equations

We study higher order nonlinear discrete Volterra equations of the form We discuss the problem of the existence of solutions with prescribed asymptotic behavior. In the proof of the main theorem we use the iterated remainder operator and the Schauder's type fixed point theorem. This approach allows us to control the degree of approximation. Our results are also new for the special cases of the above equation such as linear Volterra equation, ordinary difference equation and discrete Emden–Fowler type equation.