Oscillation of nonlinear second-order neutral delay differential equations

Available online at wwwisr-publicationscom/jnsa J Nonlinear Sci Appl, 0 07, 77 734 Research Article Journal Homepage: wwwtjnsacom - wwwisr-publicationscom/jnsa Oscillation of nonlinear second-order neutral delay differential equations Jiashan Yang a, Jingjing Wang b, Xuewen Qin a, Tongxing Li c,d, a School of Information and Electronic Engineering, Wuzhou University, Wuzhou, Guangxi 54300, P R China b School of Information Science & Technology, Qingdao University of Science & Technology, Qingdao, Shandong 6606, P R China c LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 76005, P R China d School of Informatics, Linyi University, Linyi, Shandong 76005, P R China Communicated by M Bohner Abstract By using a couple of Riccati substitutions, we establish several new oscillation criteria for a class of second-order nonlinear neutral delay differential equations These results complement and improve the related contributions reported in the literature c 07 All rights reserved Keywords: Oscillation, neutral differential equation, nonlinear equation, delayed argument, Riccati substitution 00 MSC: 34K Introduction The neutral differential equations find a number of applications in the natural sciences and technology For instance, they are frequently used in the study of distributed networks containing lossless transmission lines On the basis of these background details, this paper is concerned with the oscillatory properties of solutions to the second-order neutral delay differential equation [atφ z t] + qtfφ xδt = 0, where t > 0, zt = xt + ptxτt, φ u = u λ u, φ u = u β u, λ and β are positive constants Throughout, we assume that the following hypotheses are satisfied H a C [, +, 0, +, p, q C[, +, R, qt 0, and qt is not identically zero for large t; H τ, δ C [, +, R, τt t, δt t, lim t + τt = lim t + δt = +, τ δ = δ τ, τ t > 0, δ t > 0, and δt τt, where is a constant; H 3 f CR, R and fu/u L > 0 for all u 0, where L is a constant Corresponding author Email address: litongx007@63com Tongxing Li doi:0436/jnsa000539 Received 06-09-07

J S Yang, J J Wang, X W Qin, T X Li, J Nonlinear Sci Appl, 0 07, 77 734 78 By a solution of, we mean a function x C[T, +, R for some T which has the properties that z, aφ z C [T, +, R and satisfies on the interval [T, + A solution x of is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is said to be nonoscillatory Equation is called oscillatory if all its nontrivial solutions oscillate In recent years, there has been an increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions to different classes of differential equations, we refer the reader to [ 8] and the references cited therein The qualitative analysis of neutral differential equations is more difficult in comparison with that of ordinary differential equations; see, for instance, the monograph [] and the paper [7] Most oscillation criteria reported in the literature for equation and its particular cases have been obtained under the assumptions that and + a /λ tdt = + 0 pt <, 3 although the papers [6 8,, 3, 4,, 6, 8] were concerned with the oscillation of equation and its particular cases in the cases where 0 pt p 0 < +, 4 or pt, pt eventually, 5 0 pt = p 0, 6 where p 0 is a constant The objective of this paper is to establish new oscillation results for equation without imposing assumptions 3, 5, and 6 As is customary, all functional inequalities considered in the sequel are assumed to be satisfied for all t large enough Auxiliary inequalities To prove the main results, we need the following auxiliary inequalities Lemma [] If X 0, Y 0, and 0 < λ, then X λ + Y λ X + Y λ Lemma [7] If X 0, Y 0, and λ, then X λ + Y λ λ X + Y λ Lemma 3 [3] If B > 0 and α > 0, then 3 Oscillation criteria To present the main results, we use the notation { L, 0 < β, L 0 = β L, β >, Au Bu α+ α α α α + α+ A α+ B α ϕ +t = max{0, ϕ t}, At = where the meaning of ϕ will be explained later Qt = min{qt, qτt}, a /λ sds,

J S Yang, J J Wang, X W Qin, T X Li, J Nonlinear Sci Appl, 0 07, 77 734 79 Theorem 3 Assume that conditions H -H 3,, and 4 are satisfied If there exists a function ϕ C [, +, 0, + such that, for all constants η > 0, [ t lim sup ϕs L 0 Qs + pβ 0 t + λ/β λ aδs λ + λ+ η β λ δ s λ ] + s λ+ ds = +, λ β 3 ϕs or [ t lim sup ϕs L 0 Qs + pβ 0 η λ β/λ a β/λ δs t + β + β+ δ s β ] + s β+ ds = +, λ > β, 3 ϕs then equation is oscillatory Proof Let x be a nonoscillatory solution of equation Without loss of generality, we may assume that there exists a t such that xt > 0, xτt > 0, and xδt > 0 for t t Then, using the definition of z, we have zt > 0 and zt xt, t t From, we have [atφ z t] Lqtx β δt 0 33 By and 33, one can easily obtain z t > 0 It follows from 33 that Combining 33 and 34, we conclude that [aτtφ z τt] τ + Lqτtx β δτt 0 34 t [atφ z t] + p β 0 [aτtφ z τt] τ + Lqtx β δt + p β 0 t Lqτtxβ δτt 0 35 Assume first tha < β Applications of conditions τ t > 0, τ δ = δ τ, zt xt + p 0 xτt, 35, and Lemma yield [atφ z t] + pβ 0 [aτtφ z τt] Lqtx β δt p β 0 τ Lqτtxβ δτt 0 [ ] LQt x β δt + p β 0 xβ δτt LQt [xδt + p 0 xτδt] β LQtz β δt 36 Suppose now that β > Similarly, in view of Lemma, we have [atφ z t] + pβ [ ] 0 [aτtφ z τt] LQt x β δt + p β 0 τ xβ δτt 0 β LQt [xδt + p 0 xτδt] β β LQtz β δt 37 It follows now from 36 and 37 that [atφ z t] + pβ 0 [aτtφ z τt] L 0 Qtz β δt 38

J S Yang, J J Wang, X W Qin, T X Li, J Nonlinear Sci Appl, 0 07, 77 734 730 Case i: λ β For t t, we define a Riccati substitution wt = atφ z t φ zδt = atz t λ z β 39 δt Then wt > 0 for t t Differentiating 39, we get w t = ϕ t atz t λ z β δt + [atz t λ ] z β δt βatz t λ z β δtz δtδ t z β δt = z β δt [atz t λ ] + ϕ t wt tz t λ z δt βatδ z β+ δt 30 From 33 and z t > 0, we have aδtz δt λ atz t λ, ie, z δt at /λ z t 3 aδt Since zt > 0 and z t > 0, there exists a constant η > 0 such that zt zδt η Combining 39-3 and using Lemma 3, we obtain w t z β δt [atz t λ ] + ϕ t wt tz t λ at /λ βatδ z β+ z t δt aδt = z β δt [atz t λ ] + ϕ t wt tz β λ/λ δt βδ [aδt] /λ z β δt [atz t λ ] + ϕ +t z β δt [atz t λ ] + wt β ηβ λ/λ δ t λ λ aδt λ + λ+ β λ η β λ δ t λ Similarly, for t t, we introduce another Riccati transformation wλ+/λ t [aδt] /λ wλ+/λ t + t λ+ 3 vt = aτtφ z τt φ zδt = aτtz τt λ z β 33 δt Then vt > 0 for t t Differentiating 33, by aδtz δt λ aτtz τt λ > 0 and zδt η, we get v t = ϕ t aτtz τt λ z β δt + [aτtz τt λ ] z β δt βaτtz τt λ z β δtz δtδ t z β δt z β δt [aτtz τt λ ] + ϕ t vt β aτtz τt λ δ t z β+ δt aτt aδt z β δt [aτtz τt λ ] + ϕ +t z β δt [aτtz τt λ ] + /λ z τt vt β ηβ λ/λ δ t λ λ aδt λ + λ+ β λ η β λ δ t λ [aδt] /λ vλ+/λ t + t λ+ 34

J S Yang, J J Wang, X W Qin, T X Li, J Nonlinear Sci Appl, 0 07, 77 734 73 By virtue of 38, 3, and 34, we deduce that w t + pβ 0 v t z β δt [atz t λ ] + pβ 0 [aτtz τt λ ] + pβ 0 λ λ aδt + λ + λ+ β λ η β λ δ t λ L 0 Qt + + t λ+ + pβ 0 λ λ aδt λ + λ+ β λ η β λ δ t λ + t λ+ Integrating the latter inequality from t to t, we see that [ t ϕs L 0 Qs + pβ 0 λ/β λ aδs ϕ ] + s λ+ t λ + λ+ η β λ ds wt + pβ 0 vt, δ s λ ϕs which contradicts our assumption 3 Case ii: λ > β Define the function w by 39 Then 30 holds Using 33 and z t > 0, we have 3 Furthermore, there exists a constant η > 0 such that atz t λ aτtz τt λ η, ie, z t λ β/β aλ β/βλ t η λ β/βλ and By virtue of 39-3 and 35, we conclude that z τt λ β/β aλ β/βλ τt η λ β/βλ 35 w t z β δt [atz t λ ] + ϕ t wt tz t λ at /λ βatδ z β+ z t δt aδt z β δt [atz t λ ] + ϕ +t wt β δ twt β+/β ϕ /β ta /λ δta λ β/βλ tz t λ β/β z β δt [atz t λ ] + ϕ +t wt β δ twt β+/β ϕ /β ta /λ δtη λ β/βλ z β δt [atz t λ ] + ηλ β/λ a β/λ δt β + β+ δ t β + t β+ 36 On the other hand, define the function v as in 33 Similarly, we have v t z β δt [aτtz τt λ ] + ηλ β/λ a β/λ δt β + β+ δ t β + t β+ 37 Applications of 38, 36, and 37 imply that w t + pβ 0 v t z β [atz t λ ] + pβ 0 [aτtz τt λ ] δt + + pβ 0 η λ β/λ a β/λ δt ϕ + t β+ β + β+ δ t β L 0 Qt + + pβ 0 η λ β/λ a β/λ δt β + β+ δ t β + t β+

J S Yang, J J Wang, X W Qin, T X Li, J Nonlinear Sci Appl, 0 07, 77 734 73 Consequently, t ϕs [ L 0 Qs + pβ 0 η λ β/λ a β/λ δs β + β+ δ s β which contradicts our assumption 3 The proof is complete Choosing = Aδt, by Theorem 3, we have the following results lim sup t + ] + s β+ ds wt + pβ 0 vt, ϕs Corollary 3 Let β = λ Assume that conditions H -H 3,, and 4 are satisfied If [ L 0 AδsQs then equation is oscillatory + p λ 0 /δ s λ + λ+ A λ δsa /λ δs Corollary 33 Let β = λ Assume that conditions H -H 3,, and 4 are satisfied If or lim inf t + A λ δt lim inf t + then equation is oscillatory Proof t ln Aδt AδsQsds > AδsQsds > + p 0/ 4L Case i: λ < Condition 39 yields existence of a constant ε > 0 such that that is, A λ δt AδsQsds AδsQsds ] ds = +, 38 + p λ 0 / Lλ + λ+ λ, λ < 39 + p λ 0 / Lλ + λ+ λ + ε, + p λ 0 / Lλ + λ+ λ A λ δt εa λ δt, which implies that [ + p λ 0 AδsQs /δ ] s Lλ + λ+ A λ δsa /λ ds δs = AδsQsds εa λ δt +, λ =, 30 + p λ 0 / + p λ Lλ + λ+ λ A λ 0 δt + / Lλ + λ+ λ A λ δ + p λ 0 / Lλ + λ+ λ A λ δ + An application of Corollary 3 yields oscillation of equation as t + Case ii: λ = Note that [ln Aδs] = δ s/aδsaδs Similar as in the proof of case i, an application of condition 30 implies that 38 is satisfied By Corollary 3, equation is oscillatory This completes the proof Remark 34 Define the Riccati substitutions wt = atφ z t φ zτt and vt = aτtφ z τt φ zτt Proceeding similarly as above, one can easily establish oscillation results for equation in the case when δt τt

J S Yang, J J Wang, X W Qin, T X Li, J Nonlinear Sci Appl, 0 07, 77 734 733 Remark 35 The results obtained in this paper complement those reported in [6, 8,, 3,, 6, 8] since our criteria can be applied to the case where β λ and do not require assumptions 3, 5, and 6 Furthermore, our criteria solve a problem posed in [7, Summary] because these results can be applied to equation for β > λ 4 Example Example 4 For t and α > 0, consider the second-order neutral delay differential equation t + ϕs xt + 9 t 0 x + α t 4 t x = 0 4 5 Let β = λ =, at =, pt = p 0 = 9/0, τt = t/4, qt = α/t, δt = t/5, fu = u, L = L 0 =, and = t Then = /4, Qt = min{qt, qτt} = α/t, and so [ t lim sup L 0 Qs + pβ 0 λ/β λ aδs ϕ ] + s λ+ λ + λ+ η β λ δ s λ ds ϕs = α 3 4 lim sup t + ds = +, s provided that α > 3/4 = 575 Hence, by Theorem 3, equation 4 is oscillatory for any α > 575, whereas applications of [7, Corollary ] and [5, Theorem ] let ρt = t yield oscillation of 4 for any α > 3/e ln 5 379 and α > 5, respectively Hence, Theorem 3 improves [7, Corollary ] and [5, Theorem ] in some cases Acknowledgment This research is supported by NNSF of P R China Grant Nos 65037, 6304, and 640306, CPSF Grant Nos 05M5809 and 04M55905, NSF of Shandong Province Grant No ZR06JL0, Scientific Research Fund of Education Department of Guangxi Zhuang Autonomous Region Grant No 03YB3, DSRF of Linyi University Grant No LYDX05BS00, and the AMEP of Linyi University, P R China References [] R P Agarwal, M Bohner, W-T Li, Nonoscillation and oscillation: theory for functional differential equations, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc, New York, 004 [] R P Agarwal, M Bohner, T-X Li, C-H Zhang, A new approach in the study of oscillatory behavior of even-order neutral delay differential equations, Appl Math Comput, 5 03, 787 794 [3] R P Agarwal, M Bohner, T-X Li, C-H Zhang, Oscillation of second-order differential equations with a sublinear neutral term, Carpathian J Math, 30 04, 6 [4] R P Agarwal, M Bohner, T-X Li, C-H Zhang, Oscillation of second-order Emden Fowler neutral delay differential equations, Ann Mat Pura Appl, 93 04, 86 875 [5] R P Agarwal, C-H Zhang, T-X Li, Some remarks on oscillation of second order neutral differential equations, Appl Math Comput, 74 06, 78 8 [6] B Baculíková, J Džurina, Oscillation theorems for second order neutral differential equations, Comput Math Appl, 6 0, 94 99, 35 [7] B Baculíková, J Džurina, Oscillation theorems for second-order nonlinear neutral differential equations, Comput Math Appl, 6 0, 447 4478, 35, 4 [8] B Baculíková, T-X Li, J Dˇzurina, Oscillation theorems for second-order superlinear neutral differential equations, Math Slovaca, 63 03, 3 34, 35 [9] Z-L Han, T-X Li, S-R Sun, Y-B Sun, Remarks on the paper [Appl Math Comput 07 009 388 396], Appl Math Comput, 5 00, 3998 4007

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