Oscillations in Difference Equations with Deviating Arguments and Variable Coefficients

G E Chatzarakis,H Péics,I P Stavroulakis

doi:10.1155/2014/902616

Abstract

New sufficient conditions for the oscillation of all solutions of difference equations with several deviating arguments and variable coefficients are presented. Examples illustrating the results are also given.

Highlights

In this paper we study the oscillation of all solutions of difference equation with several variable retarded arguments of the form m
Strong interest in (ER) is motivated by the fact that it represents a discrete analogue of the differential equation
We give an example with general retarded arguments illustrating the main result of Theorem 3

Summary

Introduction

I=1 and the (dual) difference equation with several variable advanced arguments of the form m. While (EA) represents a discrete analogue of the advanced differential equation (see [1, 2] and the references cited therein). By a solution of (ER), we mean a sequence of real numbers (x(n))n≥−w which satisfies (ER) for all n ≥ 0. By a solution of (EA), we mean a sequence of real numbers (x(n))n≥0 which satisfies (EA) for all n ≥ 1. In the last few decades, the oscillatory behavior of the solutions of difference and differential equations with several deviating arguments and variable coefficients has been studied. The authors study further (ER) and (EA) and derive new sufficient oscillation conditions These conditions are the improved and generalized discrete analogues of the oscillation conditions for the corresponding differential equations, which were studied in 1982 by Ladas and Stavroulakis [2].

Oscillation Criteria

Examples