Asymptotically polynomial solutions to difference equations of neutral type

Abstract

Asymptotic properties of solutions of difference equation of the form ∆ m (x n + u n x n+k ) = a n f (n, x σ(n) ) + b n are studied. We give sufficient conditions under which all solutions, or all solutions with polynomial growth, or all nonoscillatory solutions are asymptotically polynomial. We use a new technique which allows us to control the degree of approximation.

Nonoscillatory Solutions to Second-Order Neutral Difference Equations

“…The presented results are new even for linear equations of the type defined by Equation 1, and in the case when p n ≡ 0. The first part of the proof of Theorem 1, based on the summation method and the use of discrete Bihari type lemma, is in principle standard (see [27,28,32,33]). The second part of the proof required a new approach with the use of Lemma 3.…”

“…This problem is not easy to solve. A comparison between [27] and [28] illustrates the scale of this difficulty.…”

“…The results obtained in [27] were extended to higher-order equations in [28]. In this paper, we present generalizations in a different direction, namely to difference equations with quasi-difference of the type defined by Equation (1).…”