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

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10 pages
Some oscillation criteria are established for the second order nonlinear neutral difference equations of mixed type. Δ 2 ( x n + a x n - τ 1 ± b x n + τ 2 ) α = q n x n - σ 1 β + p n x x + σ 2 β , n ≥ n 0 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 .
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Thandapaniet al.Advances in Difference Equations2012,2012:4 http://www.advancesindifferenceequations.com/content/2012/1/4
R E S E A R C HOpen Access Oscillation criteria for secondorder nonlinear neutral difference equations of mixed type 1* 12 Ethiraju Thandapani, Nagabhushanam Kavithaand Sandra Pinelas
* Correspondence: ethandapani@yahoo.co.in 1 Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India Full list of author information is available at the end of the article
Abstract Some oscillation criteria are established for the second order nonlinear neutral difference equations of mixed type. 2βα β (x+ax±bx) =q x+pnx,nn n nτ1n+τ2n nσ1x+σ20
whereaandbare ratio of odd positive integers withb1. 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. Keywords:Neutral difference equation, mixed type, comparison theorems, oscillation.
1 Introduction In this article, we study the oscillation behavior of solutions of mixed type neutral dif ference equation of the form,
2α ββ (xn+axnτ1±bxn+τ2) =qnx+pnx nσ1x+σ2
wherenÎN(n0) = {n0,n0+ 1, ...},n0is a nonnegative integer,a,bare real nonnega tive constants,τ1,τ2,s1, ands2are positive integers, {qn} and {pn} are positive real sequences anda,bare ratio of odd positive integers withb1. Letθ=max{τ1,s1}. By a solution of Equation (E±) we mean a real sequence {xn} which is defined fornn0θand satisfies Equation (E±) for allnÎN(n0). A nontri vial solution of Equation (E±) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is known as nonoscillatory. Equations of this type arise in a number of important applications such as problems in population dynamics when maturation and gestation are included, in cobweb mod els, in economics where demand depends on the price at an earlier time and in electric networks containing lossless transmission lines. Hence it is important and useful to study the oscillation behavior of solutions of neutral type difference Equation (E±). The oscillation, nonoscillation and asymptotic behavior of solutions of Equation (E±), whenb= 0 andpn0 ora= 0 andpn0 orb= 0 andqn0 have been considered by many authors, see for example [14] and the reference cited therein. However, there are few results available in the literature regarding the oscillatory properties of neutral difference equations of mixed type, see for example [18]. Motivated by the above
© 2012 Thandapani et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.