Approximative solutions to difference equations of neutral type

摘要

Asymptotic properties of solutions to difference equations of the form Δm(xn−unxn−k)=anf(xn)+bnare studied. Replacing the sequence u by its limit and the right side of the equation by zero we obtain an equation which we call the fundamental equation. First we investigate the space of all solutions of the fundamental equation. We show that any such solution is a sum of a polynomial sequence and a product of a geometric sequence and a periodic sequence. Next, using a new version of the Krasnoselski fixed point theorem and the iterated remainder operator, we establish sufficient conditions under which a given solution of the fundamental equation is an approximative solution to the above equation. Our approach, based on the iterated remainder operator, allows us to control the degree of approximation. In this paper we use o(ns), for a given nonpositive real s, as a measure of approximation.