This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Oscillation criteria for second-order nonlinear neutral difference equations of mixed type Advances in Difference Equations 2012, 2012:4 doi:10.1186/1687-1847-2012-4 Ethiraju Thandapani (ethandapani@yahoo.co.in) Nagabhushanam Kavitha (kavitha_snd@hotmail.com) Sandra Pinelas (sandra.pinelas@gmail.com) ISSN 1687-1847 Article type Research Submission date 3 October 2011 Acceptance date 27 January 2012 Publication date 27 January 2012 Article URL http://www.advancesindifferenceequations.com/content/2012/1/4 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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Oscillation criteria for second-order nonlinear neutral difference equations of mixed type Ethiraju Thandapani ∗1 , Nagabhushanam Kavitha 1 and Sandra Pinelas 2 1 Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India 2 Departamento de Matem´atica, Universidade dos A¸cores, Ponta Delgada, Portugal ∗ Corresponding author: ethandapani@yahoo.co.in Email addresses: NK: kavitha snd@hotmail.com SP: sandra.pinelas@gmail.com Abstract Some oscillation criteria are established for the second order nonlinear 1 neutral difference equations of mixed type. ∆ 2 (x n + ax n−τ 1 ± bx n+τ 2 ) α = q n x β n−σ 1 + p n x β n+σ 2 , 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. Keywords: Neutral difference equation; mixed type; comparison the- orems; oscillation. 1 Introduction In this article, we study the oscillation behavior of solutions of mixed type neutral difference equation of the form, ∆ 2 (x n + ax n−τ 1 ± bx n+τ 2 ) α = q n x β n−σ 1 + p n x β n+σ 2 (E ± ) where n ∈ N(n 0 ) = {n 0 , n 0 + 1, . . .}, n 0 is a nonnegative integer, a, b are real nonnegative constants, τ 1 , τ 2 , σ 1 , and σ 2 are positive integers, {q n } and {p n } are positive real sequences and α, β are ratio of odd positive integers with β ≥ 1. 2 Let θ = max {τ 1 , σ 1 }. By a solution of Equation (E ± ) we mean a real sequence {x n } which is defined for n ≥ n 0 −θ and satisfies Equation (E ± ) for all n ∈ N(n 0 ). A nontrivial 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 in- cluded, in cobweb models, 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 ± ), when b = 0 and p n ≡ 0 or a = 0 and p n ≡ 0 or b = 0 and q 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 observation, in this article we establish some new oscillation criteria for the Equation (E ± ) which generalize some of the results obtained in [1–3,5–7]. In Section 2, we present conditions for the oscillation of all solutions of equation (E ± ). Examples are provided in Section 3 to illustrate the results. 3 2 Oscillation results In this section, we obtain sufficient conditions for the oscillation of all solu- tions of Equation (E ± ). First we consider the Equation (E − ), viz, ∆ 2 (x n + ax n−τ 1 − bx n+τ 2 ) α = q n x β n−σ 1 + p n x β n+σ 2 , n ∈ N(n 0 ). (E − ) To prove our main results we need the following lemma, which can be found in [9]. Lemma 2.1. Let A≥0, B≥0 and γ ≥1. Then A γ + B γ ≥ 1 2 γ−1 (A + B) γ , (2.1) and A γ − B γ ≥ (A − B) γ , if A ≥ B. (2.2) Theorem 2.2. Let σ 1 > τ 1, σ 2 > τ 2 and {q n } and {p n } are positive real nonincreasing sequences. Assume that the difference inequalities i) ∆ 2 y n − p n 2 β−1 (1 + a β ) β/α y β/α n+σ 2 ≥ 0 (2.3) has no eventually positive increasing solution, ii) ∆ 2 y n − q n 2 β−1 (1 + a β ) β/α y β/α n−σ 1 +τ 1 ≥ 0 (2.4) has no eventually positive decreasing solution, 4 iii) ∆ 2 y n + q n b β y β/α n−σ 1 −τ 2 + p n b β y β/α n+σ 2 −τ 2 ≤ 0 (2.5) has no eventually positive solution. Then every solution of Equation (E − ) is oscillatory. Proof. Let {x n } be a nonoscillatory solution of Equation (E − ). Without loss of generality, we may assume that there exists n 1 ∈ N(n 0 ) such that x n−θ > 0 for all n ≥ n 1 . Set z n = (x n + ax n−τ 1 − bx n+τ 2 ) α . Then ∆ 2 z n = q n x β n−σ 1 + p n x β n+σ 2 > 0, n ≥ n 1 , which implies that {z n } and {∆z n } are of one sign for all n 2 ≥ n 1 . We claim that z n > 0 eventually. To prove it assume that z n < 0. Then we let 0 < u n = −z n = (bx n+τ 2 − ax n−τ 1 − x n ) α ≤ b α x α n+τ 2 . Thus x β n ≥ 1 b β u β/α n−τ 2 , n ≥ n 2 . From Equation (E − ), we get 0 = ∆ 2 u n + q n x β n−σ 1 + p n x β n+σ 2 ≥ ∆ 2 u n + q n b β u β/α n−σ 1 −τ 2 + p n b β u β/α n+σ 2 −τ 2 . Hence {u n } is a positive solution of inequality (2.5), a contradiction. Therefore z n ≥ 0. We define y n = z n + a β z n−τ 1 − b β 2 β−1 z n+τ 2 . (2.6) 5 Then, we have ∆ 2 y n = ∆ 2 z n + a β ∆ 2 z n−τ 1 − b β 2 β−1 ∆ 2 z n+τ 2 = q n x β n−σ 1 + p n x β n+σ 2 + a β q n−τ 1 x β n−σ 1 −τ 1 + a β p n−τ 1 x β n+σ 2 −τ 1 − b β 2 β−1 q n+τ 2 x β n−σ 1 +τ 2 − b β 2 β−1 p n+τ 2 x β n+σ 2 +τ 2 . (2.7) Using the inequality (2.1) in (2.7), we obtain ∆ 2 y n ≥ q n 2 β−1 (x n−σ 1 + ax n−σ 1 −τ 1 ) β − b β 2 β−1 q n+τ 2 x β n−σ 1 +τ 2 + p n 2 β−1 (x n+σ 2 + ax n+σ 2 −τ 1 ) β − b β 2 β−1 p n+τ 2 x β n+σ 2 +τ 2 . Now using the inequality (2.2), we obtain ∆ 2 y n ≥ q n 2 β−1 z β/α n−σ 1 + p n 2 β−1 z β/α n+σ 2 > 0. (2.8) Consequently {y n } and {∆y n } are of one sign, eventually. Now we shall prove that y n > 0. If not, then let 0 < v n = −y n = b β 2 β−1 z n+τ 2 − a β z n−τ 1 − z n ≤ b β 2 β−1 z n+τ 2 . Hence z n ≥ 2 β−1 b β v n−τ 2 , and (2.8) implies 0 ≥ ∆ 2 v n + q n b β v β/α n−σ 1 −τ 2 + p n b β v β/α n+σ 2 −τ 2 . 6 We obtain that {v n } is a positive solution of inequality (2.5), a contradiction. Next we consider the following two cases: Case 1: Let ∆z n < 0 for n ≥ n 3 ≥ n 2 . We claim that ∆y n < 0 for n ≥ n 3 . If not, then we have y n > 0, ∆y n > 0 and ∆ 2 y n ≥ 0 which implies that lim n→∞ y n = ∞. On the other hand, z n > 0 , ∆z n < 0 implies that lim n→∞ z n = c < ∞. Then applying limits on both sides of (2.6) we obtain a contradiction. Thus ∆y n < 0 for n ≥ n 3. Using the monotonicity of {z n }, we now get y n−σ 1 = z n−σ 1 + a β z n−σ 1 −τ 1 − b β 2 β−1 z n−σ 1 +τ 2 ≤ (1 + a β )z n−σ 1 −τ 1 . This together with (2.8) implies ∆ 2 y n ≥ q n 2 β−1 (1 + a β ) β/α y β/α n−σ 1 +τ 1 . Thus {y n } is a positive decreasing solution of inequality (2.4), a contradiction. Case 2: Let ∆z n > 0 for n ≥ n 3. Now we consider the following two cases. Case (i): Assume that ∆y n < 0 for n ≥ n 3. Proceeding similarly as above and using the monotonicity of {z n } we obtain y n−σ 1 ≤ (1 + a β )z n−σ 1 . Then using this in (2.8) we obtain ∆ 2 y n ≥ q n 2 β−1 z β/α n−σ 1 ≥ q n 2 β−1 (1 + a β ) β/α y β/α n−σ 1 ≥ q n 2 β−1 (1 + a β ) β/α y β/α n−σ 1 +τ 1 , and again {y n } is a positive decreasing solution of inequality (2.4), a contra- diction. 7 Case (ii): Assume that ∆y n > 0 for n ≥ n 3 . Then y n+σ 2 ≤ (1 + a β )z n+σ 2 which in view of (2.8) implies ∆ 2 y n ≥ p n 2 β−1 z β/α n+σ 2 ≥ p n 2 β−1 (1 + a β ) β/α y β/α n+σ 2 , that is, (2.3) has a positive increasing solution, a contradiction. The proof is complete. Remark 2.1. Theorem 2.2 permits us to obtain various oscillation criteria for Equation (E − ). Moreover we are able to study the asymptotic properties of solutions of Equation (E − ) even if not all assumptions of Theorem 2.2 are satisfied. If the difference inequality (2.3) has an eventually positive in- creasing solution then the conclusion of Theorem 2.2 is replaced by “Every solution of Equation (E − ) is either oscillatory or |x n | → ∞ as n → ∞”. Remark 2.2. In [2, Theorem 7.6.26] , the author considered the Equation (E − ) with α = β = 1, p n ≡ p, and q n ≡ q and obtain oscillation results with (1 + a − b) > 0. Hence Theorem 2.2 generalize and improve the results of [2, Theorem 7.6.26]. Remark 2.3. Applying existing conditions sufficient for the inequalities (2.3), (2.4), and (2.5) to have no above mentioned solutions, we immediately obtain various oscillation criteria for Equation (E − ). 8 Theorem 2.3. Let σ 1 > τ 1 , σ 2 ≥ 2, and β = α. Assume that lim sup n→∞ n+σ 2 −τ 2  s=n (n + σ 2 − s − 1) p s > (1 + a α )2 α−1 , (2.9) and lim sup n→∞ n  s=n−σ 1 +τ 1 (n − s + σ 1 − τ 1 + 1) q s > (1 + a α )2 α−1 , (2.10) and that the difference inequality (2.5) has no eventually positive solution. Then every solution of Equation (E − ) is oscillatory. Proof. Conditions (2.9) and (2.10) are sufficient for the inequality (2.3) to have no increasing positive solution and for (2.4) to have no decreasing pos- itive solution, respectively (see e.g., [2, Lemma 7.6.15]). The proof then follows from Theorem 2.2. Remark 2.4. Taking into account the result of [2], we see that the absence of positive solution of (2.5) can be replaced by the assumption that for the corresponding equation ∆ 2 y n + q n b β y β/α n−σ 1 −τ 2 + p n b β y β/α n+σ 2 −τ 2 = 0 every solution of this equation are oscillatory. Next we consider the difference Equation (E + ) ∆ 2 ( x n + ax n−τ 1 + bx n+τ 2 ) α = q n x β n−σ 1 + p n x β n+σ 2 ( E + ) and present conditions for the oscillation of all solutions of Equation (E + ). 9 [...]... difference equations J Diff Equ Appl 15 , 1077–1084 (2009) [5] Agarwal, RP, Grace, SR: The oscillation of certain difference equations Math Comput Model 30, 77–83 (1999) [6] Grace, SR: Oscillation of certain difference equations of mixed type J Math Anal Appl 224, 241–254 (1998) [7] Gyori, I, Laddas, G: Oscillation Theory of Delay Differential Equations Clarendon Press, Oxford (1991) [8] Jiang, J: Oscillation of. .. Difference Equations and Inequalities, 2nd edn Marcel Dekkar, New York (2000) [2] Agarwal, RP, Bohner, M, Grace, SR, O’Regan, D: Discrete Oscillation Theory Hindawi Publ Corporation, New York (2005) [3] Agarwal, RP, Grace, SR: Oscillation theorems for certain difference equations Dyna Syst Appl 9, 299–308 (2000) [4] Migda, M, Migda, J: Oscillatory and asymptotic properties of solutions of even order neutral. .. solution, then every solution of (E+ ) is oscillatory Proof Conditions (2.21) and (2.22) are sufficient for the inequality (2.11) to have no increasing positive solution and for (2.12) to have no decreasing positive solution, respectively (see e.g., [2, Lemma 7.6.15]) The proof then follows from Theorem 2.4 Remark 2.5 When α = β = 1 , Theorem 2.5 involves result of Theorem 7.6.6 of [2] Theorem 2.6 Let β =... 0 (2.26) has no eventually positive solution, then every solution of Equation (E+ ) is oscillatory Proof Let {xn } be a nonoscillatory solution of Equation (E+ ) Without loss of generality, we assume that there exists an integer n1 ∈ N(n0 ) such that xn−θ > 0 for all n ≥ n1 Define zn and yn as in Theorem 2.4 Proceeding as in the proof of Theorem 2.4, we obtain (2.16) Next we consider the following two... then every solution of (E+ ) is oscillatory Proof Let {xn } be a nonoscillatory solution of (E+ ) Without loss of generality, we assume that there exists an integer n1 ∈ N(n0 ) such that xn−θ > 0 for all n ≥ n1 Setting zn = (xn + axn−τ1 + bxn+τ2 )α and bβ zn+τ2 2β−1 (2.13) ∆2 zn = qn xβ 1 + pn xβ 2 ≥ 0 n−σ n+σ (2.14) yn = zn + aβ zn−τ1 + Then zn > 0, yn > 0 and 10 Then {∆zn } is of one sign, eventually... coefficient pn = 0 or the condition on {pn } is violated the conclusion of the theorem may be replaced by “Every solution {xn } of equation (E± ) is oscillatory or xn → ∞ as n → ∞” Once again from the proofs, we see that if qn = 0 or condition on {qn } is violated then the conclusion of the theorems may be replaced by “Every solution {xn } of Equation (E± ) is oscillatory or xn → 0 as n → ∞” 3 Examples In... interesting to extend the results of this article to the equation ∆ (an ∆ (xn + b xn−τ1 ± c xn+τ2 )α ) = qn xβ 1 + pn xγ 2 n+σ n−σ where α, β, and γ are ratio of odd positive integers Competing interests The authors declared that they have no competing interests Authors’ contributions ET framed the problem and NK solved the problem SP modified and made necessary changes in the proof of the theorems All authors... 1 δ1 +1 , (2.27) and n+δ2 −1 lim inf n→∞ ψs+δ2 > s=n δ2 δ2 + 1 δ2 +1 (2.28) then every solution of Equation (E+ ) is oscillatory Proof It is known (see [7]) that condition (2.27) is sufficient for inequality (2.25) to have no eventually negative solution On the other hand, condition 15 (2.28) is sufficient for inequality (2.26) to have no eventually positive solution Remark 2.6 From the results presented... ∆ yn − 4α−1 1+ aα bα + α−1 2 yn−σ1 +τ1 ≥ 0 Define An = φn vn Then {vn } is a negative solution of inequality (2.25), a contradiction This completes the proof From Theorem 2.6 and the results given in [7] we have the following oscillation criteria for Equation (E+ ) Corollary 2.7 Let β = α, δ1 = σ1 − τ1 σ2 − τ2 > 0, and δ2 = > 0 2 2 Suppose that there exist two positive real sequence {φn } and {ψn }... ∆yn > 0 In view of (2.16), we have ∆2 yn+τ2 ≥ 1 ∗ β/α p z β−1 n+τ2 n+σ2 +τ2 4 (2.17) Applying the monotonicity of zn , we find yn+σ2 = zn+σ2 + aβ zn−τ1 +σ2 + bβ zn+τ2 +σ2 ≤ 2β−1 1 + aβ + bβ 2β−1 zn+τ2 +σ2 (2.18) Combining (2.17) and (2.18) we have p∗ 2 n+τ ∆2 yn+τ2 ≥ β 4β−1 b 1 + aβ + β−1 2 11 β/α y β/α n+σ2 (2.19) Thus p∗ n ∆2 yn − bβ 1 + aβ + β−1 2 4β−1 β/α y β/α n−τ2 +σ2 ≥ 0 Therefore {yn } is a . Fully formatted PDF and full text (HTML) versions will be made available soon. Oscillation criteria for second-order nonlinear neutral difference equations of mixed type Advances in Difference Equations. cited. Oscillation criteria for second-order nonlinear neutral difference equations of mixed type Ethiraju Thandapani ∗1 , Nagabhushanam Kavitha 1 and Sandra Pinelas 2 1 Ramanujan Institute for Advanced. 39A10. Keywords: Neutral difference equation; mixed type; comparison the- orems; oscillation. 1 Introduction In this article, we study the oscillation behavior of solutions of mixed type neutral difference