Asymptotic properties of solutions of third order difference equations

Janusz Migda,Magdalena Nockowska-Rosiak,Małgorzata Migda

doi:10.2298/aadm180826006m

Abstract

We consider the difference equation of the form ?(rn?(pn?xn)) = anf (x?(n)) + bn. We present sufficient conditions under which, for a given solution y of the equation ?(rn?(pn?yn)) = 0, there exists a solution x of the nonlinear equation with the asymptotic behavior xn = yn + zn, where z is a sequence convergent to zero. Our approach allows us to control the degree of approximation, i.e., the rate of convergence of the sequence We examine two types of approximation: harmonic approximation when zn = o(ns), s ? 0, and geometric approximation when zn = o(?n), ? ? (0, 1).

Highlights

Let N, R denote the set of positive integers and the set of real numbers, respectively
The purpose of this paper is to study the asymptotic behavior of solutions of equation (E), which means that we establish conditions under which, for a given solution y of the equation
If condition (2) is satisfied, x is called a solution with prescribed asymptotic behavior, and y is called an approximative solution of (E)

Summary

INTRODUCTION

Let N, R denote the set of positive integers and the set of real numbers, respectively. The purpose of this paper is to study the asymptotic behavior of solutions of equation (E), which means that we establish conditions under which, for a given solution y of the equation (1). From the point of view of asymptotic behavior, it is worth mentioning that there exist positive sequences r and p such that any solution of (1) is convergent. It is very different from case (3) for which only constant solutions are convergent. For some other asymptotic properties of solutions to third-order difference equations with quasi-differences, we refer to [3], [7], [8], [14]. We construct three examples which show how the presented theorems can be used

PRELIMINARIES

HARMONIC APPROXIMATION

GEOMETRIC APPROXIMATION