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

Thandapani Ethiraju, Kavitha Nagabhushanam, Pinelas, Pinelas Sandra

Description

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 .

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 9
• Langue: English

Extrait

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: [e-mail] 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. 2β α β (x+ax±bx) =q x+p n x,n≥n n n−τ 1 n+τ 2 n n −σ 1 x+σ 2 0

whereaandbare ratio of odd positive integers withb≥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. 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,

2 α β β (x n +ax n−τ 1 ±bx n+τ 2 ) =q n x+p n x n−σ 1 x+σ 2

wherenÎN(n 0 ) = {n 0 ,n 0 + 1, ...},n 0 is a nonnegative integer,a,bare real nonnega tive constants,τ 1 ,τ 2 ,s 1 , ands 2 are positive integers, {q n } and {p n } are positive real sequences anda,bare ratio of odd positive integers withb≥1. Letθ=max{τ 1 ,s 1 }. By a solution of Equation (E ± ) we mean a real sequence {x n } which is defined forn≥n 0 θand satisfies Equation (E ± ) for allnÎN(n 0 ). 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 andp n ≡0 ora= 0 andp n ≡0 orb= 0 andq n ≡0 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

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