Oscillation criteria for second-order nonlinear neutral difference equations of mixed type

Ethiraju Thandapani,Sandra Pinelas,Nagabhushanam Kavitha

doi:10.1186/1687-1847-2012-4

Abstract

Some oscillation criteria are established for the second order nonlinear neutral difference equations of mixed type. where α and β are ratio of odd positive integers with β ≥ 1. Results obtained here generalize some of the results given in the literature. Examples are provided to illustrate the main results. 2010 Mathematics Subject classification: 39A10.

Highlights

1 Introduction In this article, we study the oscillation behavior of solutions of mixed type neutral difference equation of the form, 2(xn + axn−τ1 ± bxn+τ2 )α = qnxβn−σ1 + pnxβx+σ2 where n Î N(n0) = {n0, n0 + 1, ...}, n0 is a nonnegative integer, a, b are real nonnegative constants, τ1, τ2, s1, and s2 are positive integers, {qn} and {pn} are positive real sequences and a, b are ratio of odd positive integers with b ≥ 1
It would be interesting to extend the results of this article to the equation an xn + b xn−τ1 ± c xn+τ2 α = qnxβn−σ1 + pnxγn+σ2 where a, b, and g are ratio of odd positive integers
Competing interests The authors declare that they have no competing interests

Summary

Introduction

We study the oscillation behavior of solutions of mixed type neutral difference equation of the form, 2(xn + axn−τ1 ± bxn+τ2 )α = qnxβn−σ1 + pnxβx+σ2 where n Î N(n0) = {n0, n0 + 1, ...}, n0 is a nonnegative integer, a, b are real nonnegative constants, τ1, τ2, s1, and s2 are positive integers, {qn} and {pn} are positive real sequences and a, b are ratio of odd positive integers with b ≥ 1. A nontrivial solution of Equation (E±) is said to be oscillatory if it is neither eventually positive nor eventually negative.

Hence zn

Assume that