On the oscillatory behavior for a certain class of third order nonlinear delay difference equations

Samir Saker,Aiman Mukheimer,Jehad O Alzabut

doi:10.14232/ejqtde.2010.1.67

Abstract

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria for a certain class of third order nonlinear delay difference equations. Our results extend and improve some previously obtained ones. An example is worked out to demonstrate the validity of the proposed results.

Highlights

The oscillation theory and asymptotic behavior of difference equations and their applications have been and still are receiving intensive attention over the last two decades
The authors studied the asymptotic behavior of the solutions and proved that if there exists a solution xn of (11) satisfying F2(xn) < 0, where F2(xn) = 2xn(∆2xn + pnxn+1) − (∆xn)2, xn is oscillatory
Where pn and qn are positive real sequences, pn is nonincreasing, m, g, h are nonnegative integers and F (x, y) = signx ≥ |x|c1 |y|c2 where c1 and c2 are nonnegative constants such that c1 + c2 > 0. They established some sufficient conditions for the existence of oscillatory solutions

Summary

Introduction

The oscillation theory and asymptotic behavior of difference equations and their applications have been and still are receiving intensive attention over the last two decades. Established some sufficient conditions for (4) to have monotonic and nonoscillatory solutions They proved that if qn > 1 for n ≥ n0 is a positive sequence (4) is oscillatory. The authors studied the asymptotic behavior of the solutions and proved that if there exists a solution xn of (11) satisfying F2(xn) < 0, where F2(xn) = 2xn(∆2xn + pnxn+1) − (∆xn), xn is oscillatory. Where pn and qn are positive real sequences, pn is nonincreasing, m, g, h are nonnegative integers and F (x, y) = signx ≥ |x|c1 |y|c2 where c1 and c2 are nonnegative constants such that c1 + c2 > 0 They established some sufficient conditions for the existence of oscillatory solutions. Following this trend, we are concerned with the oscillation and the asymptotic behavior of solutions of the nonlinear delay difference equation of form. An example is given to demonstrate the validity of the results

Some Fundamental Lemmas

Oscillation Criteria