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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 387483, 9 pages
http://dx.doi.org/10.1155/2011/387483
Research Article

Oscillation Criteria for Certain Second-Order Nonlinear Neutral Differential Equations of Mixed Type

1School of Science, University of Jinan, Jinan, Shandong 250022, China
2School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

Received 19 September 2010; Accepted 19 January 2011

Academic Editor: Josef Diblík

Copyright © 2011 Zhenlai Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Some oscillation criteria are established for the second-order nonlinear neutral differential equations of mixed type [(𝑥(𝑡)+𝑝1𝑥(𝑡𝜏1)+𝑝2𝑥(𝑡+𝜏2))𝛾]=𝑞1(𝑡)𝑥𝛾(𝑡𝜎1)+𝑞2(𝑡)𝑥𝛾(𝑡+𝜎2), 𝑡𝑡0, where 𝛾1 is a quotient of odd positive integers. Our results generalize the results given in the literature.

1. Introduction

This paper is concerned with the oscillatory behavior of the second-order nonlinear neutral differential equation of mixed type𝑥(𝑡)+𝑝1𝑥𝑡𝜏1+𝑝2𝑥𝑡+𝜏2𝛾=𝑞1(𝑡)𝑥𝛾𝑡𝜎1+𝑞2(𝑡)𝑥𝛾𝑡+𝜎2,𝑡𝑡0.(1.1)

Throughout this paper, we will assume the following conditions hold.(A1)𝑝𝑖, 𝜏𝑖, and 𝜎𝑖, 𝑖=1,2, are positive constants;(A2)𝑞𝑖𝐶([𝑡0,),[0,)), 𝑖=1,2.

By a solution of (1.1), we mean a function 𝑥𝐶([𝑇𝑥,),) for some 𝑇𝑥𝑡0 which has the property that (𝑥(𝑡)+𝑝1𝑥(𝑡𝜏1)+𝑝2𝑥(𝑡+𝜏2))𝛾𝐶2([𝑇𝑥,),) and satisfies (1.1) on [𝑇𝑥,). As is customary, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [𝑡0,), otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Neutral functional differential equations have numerous applications in electric networks. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines which rise in high speed computers where the lossless transmission lines are used to interconnect switching circuits; see [1].

Recently, many results have been obtained on oscillation of nonneutral continuous and discrete equations and neutral functional differential equations, we refer the reader to the papers [235], and the references cited therein.

Philos [2] established some Philos-type oscillation criteria for the second-order linear differential equation 𝑟(𝑡)𝑥(𝑡)+𝑞(𝑡)𝑥(𝑡)=0,𝑡𝑡0.(1.2)

In [35], the authors gave some sufficient conditions for oscillation of all solutions of second-order half-linear differential equation ||𝑥𝑟(𝑡)||(𝑡)𝛾1𝑥(𝑡)||||+𝑞(𝑡)𝑥(𝜏(𝑡))𝛾1𝑥(𝜏(𝑡))=0,𝑡𝑡0(1.3) by employing a Riccati substitution technique.

Zhang et al. [15] examined the oscillation of even-order neutral differential equation []𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))(𝑛)+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=0,𝑡𝑡0.(1.4)

Some oscillation criteria for the following second-order quasilinear neutral differential equation ||𝑧𝑟(𝑡)||(𝑡)𝛾1𝑧(𝑡)||||+𝑞(𝑡)𝑥(𝜎(𝑡))𝛾1𝑥(𝜎(𝑡))=0,for𝑧(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡)),𝑡𝑡0(1.5) were obtained by [1217].

However, there are few results regarding the oscillatory properties of neutral differential equations with mixed arguments, see the papers [2024]. In [25], the authors established some oscillation criteria for the following mixed neutral equation:𝑥(𝑡)+𝑝1𝑥𝑡𝜏1+𝑝2𝑥𝑡+𝜏2=𝑞1(𝑡)𝑥𝑡𝜎1+𝑞2(𝑡)𝑥𝑡+𝜎2,𝑡𝑡0;(1.6) here 𝑞1 and 𝑞2 are nonnegative real-valued functions. Grace [26] obtained some oscillation theorems for the odd order neutral differential equation 𝑥(𝑡)+𝑝1𝑥𝑡𝜏1+𝑝2𝑥𝑡+𝜏2(𝑛)=𝑞1𝑥𝑡𝜎1+𝑞2𝑥𝑡+𝜎2,𝑡𝑡0,(1.7) where 𝑛1 is odd. Grace [27] and Yan [28] obtained several sufficient conditions for the oscillation of solutions of higher-order neutral functional differential equation of the form (𝑥(𝑡)+𝑐𝑥(𝑡)+𝐶𝑥(𝑡+𝐻))(𝑛)+𝑞𝑥(𝑡𝑔)+𝑄𝑥(𝑡+𝐺)=0,𝑡𝑡0,(1.8) where 𝑞 and 𝑄 are nonnegative real constants.

Clearly, (1.6) is a special case of (1.1). The purpose of this paper is to study the oscillation behavior of (1.1).

In the sequel, when we write a functional inequality without specifying its domain of validity we assume that it holds for all sufficiently large 𝑡.

2. Main Results

In the following, we give our results.

Theorem 2.1. Assume that 𝜎𝑖>𝜏𝑖, 𝑖=1,2. If limsup𝑡𝑡+𝜎2𝜏2𝑡𝑡+𝜎2𝜏2𝑄𝑠22(𝑠)d𝑠>𝛾121+𝑝𝛾1+𝑝𝛾22𝛾1,(2.1)limsup𝑡𝑡𝑡𝜎1+𝜏1𝑠𝑡+𝜎1𝜏1𝑄12(𝑠)d𝑠>𝛾121+𝑝𝛾1+𝑝𝛾22𝛾1,(2.2) where 𝑄𝑖𝑞(𝑡)=min𝑖𝑡𝜏1,𝑞𝑖(𝑡),𝑞𝑖𝑡+𝜏2,(2.3) for 𝑖=1,2, then every solution of (1.1) oscillates.

Proof. Let 𝑥 be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists 𝑡1𝑡0 such that 𝑥(𝑡)>0, 𝑥(𝑡𝜏1)>0,𝑥(𝑡+𝜏2)>0, 𝑥(𝑡𝜎1)>0, and 𝑥(𝑡+𝜎2)>0 for all 𝑡𝑡1. Setting 𝑧(𝑡)=𝑥(𝑡)+𝑝1𝑥𝑡𝜏1+𝑝2𝑥𝑡+𝜏2𝛾,𝑦(𝑡)=𝑧(𝑡)+𝑝𝛾1𝑧𝑡𝜏1+𝑝𝛾22𝛾1𝑧𝑡+𝜏2.(2.4) Thus 𝑧(𝑡)>0, 𝑦(𝑡)>0, and 𝑧(𝑡)=𝑞1(𝑡)𝑥𝛾𝑡𝜎1+𝑞2(𝑡)𝑥𝛾𝑡+𝜎20.(2.5) Then, 𝑧(𝑡) is of constant sign, eventually. On the other hand, 𝑦(𝑡)=𝑞1(𝑡)𝑥𝛾𝑡𝜎1+𝑞2(𝑡)𝑥𝛾𝑡+𝜎2+𝑝𝛾1𝑞1𝑡𝜏1𝑥𝛾𝑡𝜏1𝜎1+𝑝𝛾1𝑞2𝑡𝜏1𝑥𝛾𝑡𝜏1+𝜎2+𝑝𝛾22𝛾1𝑞1𝑡+𝜏2𝑥𝛾𝑡+𝜏2𝜎1+𝑝𝛾22𝛾1𝑞2𝑡+𝜏2𝑥𝛾𝑡+𝜏2+𝜎2.(2.6) Note that 𝑔(𝑢)=𝑢𝛾, 𝛾1, 𝑢(0,) is a convex function. Hence, by the definition of convex function, we obtain 𝑎𝛾+𝑏𝛾12𝛾1(𝑎+𝑏)𝛾.(2.7) Using inequality (2.7), we get 𝑥𝛾𝑡𝜎1+𝑝𝛾1𝑥𝛾𝑡𝜏1𝜎112𝛾1𝑥𝑡𝜎1+𝑝1𝑥𝑡𝜏1𝜎1𝛾,12𝛾1𝑥𝑡𝜎1+𝑝1𝑥𝑡𝜏1𝜎1𝛾+𝑝𝛾22𝛾1𝑥𝛾𝑡+𝜏2𝜎112𝛾12𝑥𝑡𝜎1+𝑝1𝑥𝑡𝜏1𝜎1+𝑝2𝑥𝑡+𝜏2𝜎1𝛾=𝑧𝑡𝜎12𝛾12.(2.8) Similarly, we obtain 𝑥𝛾𝑡+𝜎2+𝑝𝛾1𝑥𝛾𝑡𝜏1+𝜎2+𝑝𝛾22𝛾1𝑥𝛾𝑡+𝜏2+𝜎2𝑧𝑡+𝜎22𝛾12.(2.9) Thus, from (2.6), we have 𝑦1(𝑡)2𝛾12𝑄1(𝑡)𝑧𝑡𝜎1+𝑄2(𝑡)𝑧𝑡+𝜎2.(2.10)
In the following, we consider two cases.Case 1. Assume that 𝑧(𝑡)>0. Then, 𝑦(𝑡)>0. In view of (2.10), we see that 𝑦𝑡+𝜏212𝛾12𝑄2𝑡+𝜏2𝑧𝑡+𝜏2+𝜎2.(2.11) Applying the monotonicity of 𝑧, we find 𝑦𝑡+𝜎2=𝑧𝑡+𝜎2+𝑝𝛾1𝑧𝑡𝜏1+𝜎2+𝑝𝛾22𝛾1𝑧𝑡+𝜏2+𝜎21+𝑝𝛾1+𝑝𝛾22𝛾1𝑧𝑡+𝜏2+𝜎2.(2.12) Combining the last two inequalities, we obtain the inequality 𝑦𝑡+𝜏2𝑄2𝑡+𝜏22𝛾121+𝑝𝛾1+𝑝𝛾2/2𝛾1𝑦𝑡+𝜎2.(2.13) Therefore, 𝑦 is a positive increasing solution of the differential inequality 𝑦𝑄(𝑡)2(𝑡)2𝛾121+𝑝𝛾1+𝑝𝛾2/2𝛾1𝑦𝑡𝜏2+𝜎2.(2.14) However, by [11], condition (2.1) contradicts the existence of a positive increasing solution of inequality (2.14).
Case 2. Assume that 𝑧(𝑡)<0. Then, 𝑦(𝑡)<0. In view of (2.10), we see that 𝑦𝑡𝜏112𝛾12𝑄1𝑡𝜏1𝑧𝑡𝜏1𝜎1.(2.15) Applying the monotonicity of 𝑧, we find 𝑦𝑡𝜎1=𝑧𝑡𝜎1+𝑝𝛾1𝑧𝑡𝜏1𝜎1+𝑝𝛾212𝛾1𝑧𝑡+𝜏2𝜎11+𝑝𝛾1+𝑝𝛾22𝛾1𝑧𝑡𝜏1𝜎1.(2.16) Combining the last two inequalities, we obtain the inequality 𝑦𝑡𝜏1𝑄1𝑡𝜏12𝛾121+𝑝𝛾1+𝑝𝛾2/2𝛾1𝑦𝑡𝜎1.(2.17) Therefore, 𝑦 is a positive decreasing solution of the differential inequality 𝑦𝑄(𝑡)1(𝑡)2𝛾121+𝑝𝛾1+𝑝𝛾2/2𝛾1𝑦𝑡+𝜏1𝜎1.(2.18) However, by [11], condition (2.2) contradicts the existence of a positive decreasing solution of inequality (2.18).

Remark 2.2. When 𝛾=1, Theorem 2.1 involves results of [25, Theorem 1].

Theorem 2.3. Let 𝛽𝑖=(𝜎𝑖𝜏𝑖)/2>0, 𝑖=1,2. Suppose that, for 𝑖=1,2, there exist functions 𝑎𝑖𝐶1𝑡0,,𝑎𝑖(𝑡)>0,(1)𝑖𝑎𝑖(𝑡)0,(2.19) such that 𝑄𝑖2(𝑡)𝛾121+𝑝𝛾1+𝑝𝛾22𝛾1𝑎𝑖(𝑡)𝑎𝑖𝑡+(1)𝑖𝛽𝑖,(2.20) where 𝑄𝑖 are as in (2.3) for 𝑖=1,2. If the first-order differential inequality 𝑣(𝑡)+(1)𝑖+1𝑎𝑖𝑡+(1)𝑖𝛽𝑖𝑣𝑡+(1)𝑖𝛽𝑖0(2.21) has no eventually negative solution for 𝑖=1 and no eventually positive solution for 𝑖=2, then (1.1) is oscillatory.

Proof. Let 𝑥 be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists 𝑡1𝑡0 such that 𝑥(𝑡)>0, 𝑥(𝑡𝜏1)>0, 𝑥(𝑡+𝜏2)>0, 𝑥(𝑡𝜎1)>0, and 𝑥(𝑡+𝜎2)>0 for all 𝑡𝑡1. Define 𝑧 and 𝑦 as in Theorem 2.1. Proceeding as in the proof of Theorem 2.1, we get (2.10).
In the following, we consider two cases.Case 1. Assume that 𝑧(𝑡)>0. Clearly, 𝑦(𝑡)>0. Then, just as in Case 1 of Theorem 2.1, we find that 𝑦 is a positive increasing solution of inequality (2.14). Let 𝑏2(𝑡)=𝑦(𝑡)+𝑎2(𝑡)𝑦(𝑡+𝛽2). Then 𝑏2(𝑡)>0. Using (2.19) and (2.20), we obtain 𝑏2𝑎(𝑡)2(𝑡)𝑎2𝑏(𝑡)2(𝑡)𝑎2(𝑡)𝑏2𝑡+𝛽2=𝑦𝑎(𝑡)2(𝑡)𝑎2𝑦(𝑡)(𝑡)𝑎2(𝑡)𝑎2𝑡+𝛽2𝑦𝑡+2𝛽2𝑦(𝑡)𝑎2(𝑡)𝑎2𝑡+𝛽2𝑦𝑡+2𝛽2𝑦𝑄(𝑡)2(𝑡)2𝛾121+𝑝𝛾1+𝑝𝛾2/2𝛾1𝑦𝑡𝜏2+𝜎20.(2.22) Define 𝑏2(𝑡)=𝑎2(𝑡)𝑣(𝑡). Then, 𝑣 is a positive solution of (2.21) for 𝑖=2, which is a contradiction.
Case 2. Assume that 𝑧(𝑡)<0. Clearly, 𝑦(𝑡)<0. Then, just as in Case 2 of Theorem 2.1, we find that 𝑦 is a positive decreasing solution of inequality (2.18). Let 𝑏1(𝑡)=𝑦(𝑡)𝑎1(𝑡)𝑦(𝑡𝛽1). Then 𝑏1(𝑡)<0. Using (2.19) and (2.20), we obtain 𝑏1𝑎(𝑡)1(𝑡)𝑎1𝑏(𝑡)1(𝑡)+𝑎1(𝑡)𝑏1𝑡𝛽1=𝑦𝑎(𝑡)1(𝑡)𝑎1𝑦(𝑡)(𝑡)𝑎1(𝑡)𝑎1𝑡𝛽1𝑦𝑡2𝛽1𝑦(𝑡)𝑎1(𝑡)𝑎1𝑡𝛽1𝑦𝑡2𝛽1𝑦𝑄(𝑡)1(𝑡)2𝛾121+𝑝𝛾1+𝑝𝛾2/2𝛾1𝑦𝑡+𝜏1𝜎10.(2.23) Define 𝑏1(𝑡)=𝑎1(𝑡)𝑣(𝑡). Then, 𝑣 is a negative solution of (2.21) for 𝑖=1. This contradiction completes the proof of the theorem.

Remark 2.4. When 𝛾=1, Theorem 2.3 involves results of [25, Theorem 2].

From Theorem 2.3 and the results given in [12], we have the following oscillation criterion for (1.1).

Corollary 2.5. Let 𝛽𝑖=(𝜎𝑖𝜏𝑖)/2>0, 𝑖=1,2. Assume that (2.19) and (2.20) hold for 𝑖=1,2. If liminf𝑡𝑡𝑡𝛽1𝑎1𝑠𝛽11d𝑠>e,(2.24)liminf𝑡𝑡+𝛽2𝑡𝑎2𝑠+𝛽21d𝑠>e,(2.25) then (1.1) is oscillatory.

Proof. It is known (see [12]) that condition (2.24) is sufficient for inequality (2.21) (for 𝑖=1) to have no eventually negative solution. On the other hand, condition (2.25) is sufficient for inequality (2.21) (for 𝑖=2) to have no eventually positive solution.

For an application of our results, we give the following example.

Example 2.6. Consider the second-order differential equation 𝑥(𝑡)+𝑝1𝑥𝑡𝜏1+𝑝2𝑥𝑡+𝜏2𝛾=𝑞1𝑥𝛾𝑡𝜎1+𝑞2𝑥𝛾𝑡+𝜎2,𝑡𝑡0,(2.26) where 𝑞𝑖>0 are constants and 𝜎𝑖>𝜏𝑖 for 𝑖=1,2.
It is easy to see that 𝑄𝑖(𝑡)=𝑞𝑖, 𝑖=1,2. Assume that 𝜀>0. Let 𝑎𝑖(𝑡)=(2+𝜀)/(e(𝜎𝑖𝜏𝑖)), 𝑖=1,2. Clearly, (2.19) holds. If 𝑞𝑖>2e𝜎𝑖𝜏𝑖22𝛾121+𝑝𝛾1+𝑝𝛾22𝛾1(2.27) for 𝑖=1,2, then (2.20) holds. Moreover, we see that liminf𝑡𝑡𝑡𝛽1𝑎1𝑠𝛽1d𝑠=2+𝜀>12ee,liminf𝑡𝑡+𝛽2𝑡𝑎2𝑠+𝛽2d𝑠=2+𝜀>12ee.(2.28) Hence by applying Corollary 2.5, we find that (2.26) is oscillatory.

Acknowledgments

The authors sincerely thank the referees for their constructive suggestions which improved the content of the paper. This research is supported by the Natural Science Foundation of China (11071143, 60904024, 11026112), China Postdoctoral Science Foundation funded project (200902564), and Shandong Provincial Natural Science Foundation (ZR2010AL002, ZR2009AL003, Y2008A28); it was also supported by University of Jinan Research Funds for Doctors (XBS0843).

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