About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2011 (2011), Article ID 367541, 17 pages
http://dx.doi.org/10.1155/2011/367541
Research Article

Oscillation Criteria for Second-Order Superlinear Neutral Differential Equations

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

Received 5 September 2010; Accepted 20 January 2011

Academic Editor: Josef Diblík

Copyright © 2011 Tongxing Li 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 superlinear neutral differential equations (??(??)|??'(??)|??-1??'(??))'+??(??,??(??(??)))=0, ??=??0, where ??(??)=??(??)+??(??)??(??(??)), ??(??)=??, ??(??)=??, ?????([??0,8),[0,??0]), and ??=1. Our results are based on the cases ?8??01/??1/??(??)d??=8 or ?8??01/??1/??(??)d??<8. Two examples are also provided to illustrate these results.

1. Introduction

This paper is concerned with the oscillatory behavior of the second-order superlinear differential equation?||????(??)?||(??)??-1????(??)?+??(??,??(??(??)))=0,??=??0,(1.1) where ??=1 is a constant, ??(??)=??(??)+??(??)??(??(??)).

Throughout this paper, we will assume the following hypotheses:(??1)?????1([??0,8),R),??(??)>0 for ??=??0;(??2)?????([??0,8),[0,??0]), where ??0 is a constant;(??3)?????1([??0,8),R), ??'(??)=??0>0, ??(??)=??; (??4)?????([??0,8),R), ??(??)=??,??°??=??°??; (??5)??(??,??)???([??0,8)×R,R), and there exists a function ?????([??0,8),[0,8)) such that ??(??,??)sign??=??(??)|??|??,for???0,??=??0.(1.2)

By a solution of (1.1), we mean a function ?????([????,8),R) for some ????=??0 which has the property that ??(??)|??'(??)|??-1??'(??)???1([????,8),R) and satisfies (1.1) on [????,8). We consider only those solutions ?? which satisfy sup{|??(??)|:??=??}>0 for all ??=????. As is customary, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [??0,8), otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

We note that neutral differential equations find 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].

In the last few years, there are many studies that have been made on the oscillation and asymptotic behavior of solutions of discrete and continuous equations; see, for example, [228]. Agarwal et al. [5], Chern et al. [6], Džurina and Stavroulakis [7], Kusano and Yoshida [8], Kusano and Naito [9], Mirzov [10], and Sun and Meng [11] observed some similar properties between?||????(??)?||(??)??-1????(??)?||||+??(??)??(??(??))??-1??(??(??))=0(1.3) and the corresponding linear equations???(??)????(??)?+??(??)??(??)=0.(1.4) Baculíková [12] established some new oscillation results for (1.3) when ??=1. In 1989, Philos [13] obtained some Philos-type oscillation criteria for (1.4).

Recently, many results have been obtained on oscillation and nonoscillation of neutral differential equations, and we refer the reader to [1435] and the references cited therein. Liu and Bai [16], Xu and Meng [17, 18], Dong [19], Baculíková and Lacková [20], and Jiang and Li [21] established some oscillation criteria for (1.3) with neutral term under the assumptions ??(??)=??, ??(??)=??,???(??)=????01??1/???(??)d???8as???8,(1.5)8??01??1/??(??)d??<8.(1.6)

Saker and O'Regan [24] studied the oscillatory behavior of (1.1) when 0=??(??)<1, ??(??)=?? and ??(??)>??.

Han et al. [26] examined the oscillation of second-order nonlinear neutral differential equation?[]??(??)??(??)+??(??)??(??(??))???+??(??)??(??(??(??)))=0,??=??0,(1.7) where ??(??)=??, ??(??)=??, ??'(??)=??0>0, 0=??(??)=??0<8, and the authors obtained some oscillation criteria for (1.7).

However, there are few results regarding the oscillatory problem of (1.1) when ??(??)=?? and ??(??)=??. Our aim in this paper is to establish some oscillation criteria for (1.1) under the case when ??(??)=?? and ??(??)=??.

The paper is organized as follows. In Section 2, we will establish an inequality to prove our results. In Section 3, some oscillation criteria are obtained for (1.1). In Section 4, we give two examples to show the importance of the main results.

All functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all ?? large enough.

2. Lemma

In this section, we give the following lemma, which we will use in the proofs of our main results.

Lemma 2.1. Assume that ??=1,????,???R. If ??=0,????=0, then ????+????=12??-1(??+??)??(2.1) holds.

Proof. (i) Suppose that ??=0 or ??=0. Obviously, we have (2.1). (ii) Suppose that ??>0, ??>0. Define the function ?? by ??(??)=????,?????(0,8). Then ????(??)=??(??-1)????-2=0 for ??>0. Thus, ?? is a convex function. By the definition of convex function, for ??=1/2,??,???(0,8), we have ?????+??2?=??(??)+??(??)2,(2.2) that is, ????+????=12??-1(??+??)??.(2.3) This completes the proof.

3. Main Results

In this section, we will establish some oscillation criteria for (1.1).

First, we establish two comparison theorems which enable us to reduce the problem of the oscillation of (1.1) to the research of the first-order differential inequalities.

Theorem 3.1. Suppose that (1.5) holds. If the first-order neutral differential inequality ??????(??)+0?????0???(??(??))?+12??-1??????(??)??(??(??))-??1??????(??(??))=0(3.1) has no positive solution for all sufficiently large ??1, where ??(??)=min{??(??),??(??(??))}, then every solution of (1.1) oscillates.

Proof. Let ?? be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists ??1=??0 such that ??(??)>0,????(??(??))>0, and ??(??(??))>0 for all ??=??1. Then ??(??)>0 for ??=??1. In view of (1.1), we obtain ?||????(??)?||(??)??-1????(??)?=-??(??)????(??(??))=0,??=??1.(3.2) Thus, ??(??)|???(??)|??-1??'(??) is decreasing function. Now we have two possible cases for ???(??): (i) ???(??)<0 eventually, (ii) ??'(??)>0 eventually.
Suppose that ??'(??)<0 for ??=??2=??1. Then, from (3.2), we get ||????(??)?||(??)??-1??????(??)=??2?||??????2?||??-1??????2?,??=??2,(3.3) which implies that ?????(??)=??2?-??1/?????2?||??????2?||?????2??-1/??(??)d??.(3.4) Letting ???8, by (1.5), we find ??(??)?-8, which is a contradiction. Thus ???(??)>0(3.5) for ??=??2.
By applying (1.1), for all sufficiently large ??, we obtain ??????(??)?(???)????+??(??)????(?????(??))+0?????(??(??))????(?????(??(??)))+0??????????(??)??(??(??))?(???(??))????=0.(3.6) Using inequality (2.1), (3.2), (3.5), ??°??=??°??, and the definition of ??, we conclude that ??????(??)?(???)????+???0?????0?????(??(??))?(??(??))????+12??-1??(??)????(??(??))=0.(3.7) It follows from (3.2) and (3.5) that ??(??)=??(??)(???(??))??>0 is decreasing and then ???(??)=????2??????(??)??(??)???1/????1/??(??)d??=??1/???(??)????21??1/??(??)d??=??1/??????(??)??(??)-??2??.(3.8) Thus, from (3.7) and the above inequality, we find ??????(??)+0?????0???(??(??))?+12??-1??????(??)??(??(??))-??2??????(??(??))=0.(3.9) That is, inequality (3.1) has a positive solution ??; this is a contradiction. The proof is complete.

Theorem 3.2. Suppose that (1.5) holds. If the first-order differential inequality ?????(??)+02??-1???0+???0??????????(??)??(??(??))-??1??????(??(??))=0(3.10) has no positive solution for all sufficiently large ??1, where ?? is defined as in Theorem 3.1, then every solution of (1.1) oscillates.

Proof. Let ?? be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists ??1=??0 such that ??(??)>0,????(??(??))>0, and ??(??(??))>0 for all ??=??1. Then ??(??)>0 for ??=??1. Proceeding as in the proof of Theorem 3.1, we obtain that ??(??)=??(??)(??'(??))?? is decreasing, and it satisfies inequality (3.1). Set ??(??)=??(??)+(??0)????(??(??))/??0. From ??(??)=??, we get ?????(??)=??(??)+0?????0??????(??(??))=1+0?????0???(??).(3.11) In view of the above inequality and (3.1), we see that ?????(??)+02??-1???0+???0??????????(??)??(??(??))-??1??????(??(??))=0.(3.12) That is, inequality (3.10) has a positive solution ??; this is a contradiction. The proof is complete.

Next, using Riccati transformation technique, we obtain the following results.

Theorem 3.3. Suppose that (1.5) holds. Moreover, assume that there exists ?????1([??0,8),(0,8)) such that limsup???8?????0???(??)??(??)2??-1-1(??+1)??+1????1+0?????0??????(??)?+?(??)??+1(??(??))???d??=8(3.13) holds, where ?? is defined as in Theorem 3.1, ???+(??)=max{0,???(??)}. Then every solution of (1.1) oscillates.

Proof. Let ?? be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists ??1=??0 such that ??(??)>0, ??(??(??))>0, and ??(??(??))>0 for all ??=??1. Then ??(??)>0 for ??=??1. Proceeding as in the proof of Theorem 3.1, we obtain (3.2)–(3.7).
Define a Riccati substitution ?????(??)=??(??)??(??)??(??)??(??(??))??,??=??2.(3.14) Thus ??(??)>0 for ??=??2. Differentiating (3.14) we find that ???(??)=????????(??)(??)??(??)??(??(??))????????+??(??)(??)??(??)????(??(??))???????-????(??)(??)??(??)??????-1(??)??'(??)(??(??))2??.(3.15) Hence, by (3.14) and (3.15), we see that ?????(??)=?(??)????????(??)??(??)+??(??)(??)??(??)????(??(??))??-????1/??(??)??1/??(????)(??+1)/??(??).(3.16)
Similarly, we introduce another Riccati substitution ?????(??)=??(??)??(??(??))??(??(??))??(??(??))??,??=??2.(3.17) Then ??(??)>0 for ??=??2. From (3.2), (3.5), and ??(??)=??, we have ????(??)=??(??(??))???(??)1/?????(??(??)).(3.18) Differentiating (3.17), we find ???(??)=????????(??)(??(??))??(??(??))??(??(??))????????+??(??)(??(??))??(??(??))????(??(??))???????-????(??)(??(??))??(??(??))??????-1(??)???(??)(??(??))2??.(3.19) Therefore, by (3.17), (3.18), and (3.19), we see that ??'(??)=??'(??)????????(??)??(??)+??(??)(??(??))??(??(??))????(??(??))??-????1/??(??)??1/??(????)(??+1)/??(??).(3.20) Thus, from (3.16) and (3.20), we have ??????(??)+0?????0?????????(??)=??(??)(??)??(??)????+????0???/??0???????(??(??))??(??(??))????(??(??))??+???(??)????(??)??(??)-??1/??(??)??1/????(??)(??+1)/??(?????)+0?????0???(??)-?????(??)??(??)0?????0????1/??(??)??1/????(??)(??+1)/??(??).(3.21) It follows from (3.5), (3.7), and ??(??)=?? that ???(?????)+0?????0???(1??)=-2??-1????(??)??(??)+?+(??)????(??)??(??)-??1/??(??)??1/????(??)(??+1)/??(+?????)0?????0???+(??)?????(??)??(??)-0?????0????1/??(??)??1/????(??)(??+1)/??(??).(3.22) Integrating the above inequality from ??2 to ??, we obtain ?????(??)-??2?+???0?????0?????(??)-0?????0?????2??=-????212??-1???(??)??(??)d??+????2????+(??)????(??)??(??)-??1/??(??)??1/????(??)(??+1)/???+?(??)d??????2???0?????0????+(??)????(??)??(??)-??1/??(??)??1/??(????)(??+1)/???(??)d??.(3.23) Define ?????:=??1/??(??)??1/???(??)??/(??+1)?????(??),??:=?+(??)????(??)?????+1??1/??(??)??1/???(??)-??/(??+1)???.(3.24) Using inequality ??+1??????1/??-??(??+1)/??=1????(??+1)/??,for??=0,??=0areconstants,(3.25) we have ???+(??)????(??)??(??)-??1/??(??)??1/????(??)(??+1)/??1(??)=(??+1)??+1?????(??)?+?(??)??+1??(??)??.(3.26) Similarly, we obtain ???+(??)????(??)??(??)-??1/??(??)??1/????(??)(??+1)/??1(??)=(??+1)??+1?????(??)?+?(??)??+1??(??)??.(3.27) Thus, from (3.23), we get ?????(??)-??2?+???0?????0?????(??)-0?????0?????2??=-????2???(??)??(??)2??-1-1(??+1)??+1????1+0?????0??????(??)?+?(??)??+1??(??)???d??,(3.28) which contradicts (3.13). This completes the proof.

As an immediate consequence of Theorem 3.3 we get the following.

Corollary 3.4. Let assumption (3.13) in Theorem 3.3 be replaced by limsup???8?????0??(??)??(??)d??=8,limsup???8?????0?????(??)?+?(??)??+1(??(??))??d??<8.(3.29) Then every solution of (1.1) oscillates.

From Theorem 3.3 by choosing the function ??, appropriately, we can obtain different sufficient conditions for oscillation of (1.1), and if we define a function ?? by ??(??)=1, and ??(??)=??, we have the following oscillation results.

Corollary 3.5. Suppose that (1.5) holds. If limsup???8?????0??(??)d??=8,(3.30) where ?? is defined as in Theorem 3.1, then every solution of (1.1) oscillates.

Corollary 3.6. Suppose that (1.5) holds. If limsup???8?????0?????(??)2??-1-1(??+1)??+1????1+0?????0???(??)?????d??=8,(3.31) where ?? is defined as in Theorem 3.1, then every solution of (1.1) oscillates.

In the following theorem, we present a Philos-type oscillation criterion for (1.1).

First, we introduce a class of functions R. Let ??0=?(??,??):??>??=??0??,??=(??,??):??=??=??0?.(3.32) The function ?????(??,R) is said to belong to the class R (defined by ???R, for short) if(i)??(??,??)=0, for ??=??0,????(??,??)>0, for (??,??)???0;(ii)?? has a continuous and nonpositive partial derivative ????(??,??)/???? on ??0 with respect to ??.

We assume that ??(??) and ??(??) for ??=??0 are given continuous functions such that ??(??)>0 and differentiable and define ????(??)=?(??)??(??)+(??+1)(??(??))1/???[],??(??)=??(??)??(??)??(??)?-??(??)(??(??))(1+??)/???,??(??,??)=??(??)??(??)(??+1)??+1???(??)+????(??,??)/???????(??,??)??+1.(3.33)

Now, we give the following result.

Theorem 3.7. Suppose that (1.5) holds and ?? is a quotient of odd positive integers. Moreover, let ???R be such that limsup???81?????,??0??????0???(??,??)??(??)??(??)2??-1-????1+0?????0??(??(??)+??(??,??))d??=8(3.34) holds, where ?? is defined as in Theorem 3.1. Then every solution of (1.1) oscillates.

Proof. Let ?? be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists ??1=??0 such that ??(??)>0,????(??(??))>0, and ??(??(??))>0 for all ??=??1. Then ??(??)>0 for ??=??1. Proceeding as in the proof of Theorem 3.1, we obtain (3.2)–(3.7). Define the Riccati substitution ?? by ??????(??)=??(??)??(??)??(??)??(??(??))???+??(??)??(??),??=??2=??1.(3.35) Then, we have ???(??)=???????(??)??(??)??(??)??(??(??))???????+??(??)??(??)+??(??)??(??)??(??)??(??(??))???+??(??)??(??)?=???(??)??[](??)??(??)+??(??)??(??)??(??)???????+??(??)(??)??(??)????(??(??))???????-????(??)(??)??(??)??+1(??(??))??+1.(3.36) Using (3.35), we get ?????(??)=?(??)??[](??)??(??)+??(??)??(??)??(??)???????+??(??)(??)??(??)????(??(??))??-????(??)??1/??(???)??(??)???(??)-??(??)??(??)(1+??)/??.(3.37) Let ??=??(??)??(??),??=??(??)??(??).(3.38) By applying the inequality (see [21, 24]) ??(1+??)/??-(??-??)1+??/??=??1/??1??1+???1??-?????,for??=oddodd=1,(3.39) we see that ???(??)???(??)-??(??)??(??)(1+??)/??=??(1+??)/??(??)??(1+??)/??+1(??)??(??(??)??(??))(1+??)/??-??+1??(??(??)??(??))1/????(??)??(??).(3.40) Substituting (3.40) into (3.37), we have ??????(??)=?(??)??(??)+(??+1)(??(??))1/????[]??(??)+??(??)??(??)??(??)?-??(??)(??(??))(1+??)/???????+??(??)??(??)??(??)????(??(??))??-????1/??(??)??1/????(??)(1+??)/??(??).(3.41) That is, ?????????(??)=??(??)??(??)+??(??)+??(??)(??)??(??)????(??(??))??-????1/??(??)??1/??(????)(1+??)/??(??).(3.42)
Next, define another Riccati transformation ?? by ??????(??)=??(??)??(??(??))??(??(??))??(??(??))???+??(??)??(??),??=??2=??1.(3.43) Then, we have ???(??)=???????(??)??(??(??))??(??(??))??(??(??))???????+??(??)??(??)+??(??)??(??(??))??(??(??))??(??(??))???+??(??)??(??)?=???(??)[]??(??)??(??)+??(??)??(??)??(??)?????+??(??)??(??(??))??(??(??))????(??(??))?????-????(??)??(??(??))??(??(??))?????(??)(??(??))??+1.(3.44) From (3.2), (3.5), and ??(??)=??, we have that (3.18) holds. Hence, we obtain ?????(??)=?(??)[]??(??)??(??)+??(??)??(??)??(??)?????+??(??)??(??(??))??(??(??))???(??(??))??????-????(??)??(??(??))??(??(??))???(1+??)/????1/??(??)(??(??))??+1.(3.45) Using (3.43), we get ?????(??)=?(??)??[](??)??(??)+??(??)??(??)??(??)???????+??(??)(??(??))??(??(??))????(??(??))??-????(??)??1/??(???)??(??)???(??)-??(??)??(??)(1+??)/??.(3.46) Let ??=??(??)??(??),??=??(??)??(??).(3.47) By applying the inequality (3.39), we see that ???(??)???(??)-??(??)??(??)(1+??)/??=??(1+??)/??(??)??(1+??)/??+1(??)??(??(??)??(??))(1+??)/??-??+1??(??(??)??(??))1/????(??)??(??).(3.48) Substituting (3.48) into (3.46), we have ??????(??)=?(??)??(??)+(??+1)(??(??))1/????[]??(??)+??(??)??(??)??(??)?-??(??)(??(??))(1+??)/???????+??(??)??(??(??))??(??(??))????(??(??))??-????1/??(??)??1/????(??)(1+??)/??(??).(3.49) That is, ?????????(??)=??(??)??(??)+??(??)+??(??)(??(??))??(??(??))????(??(??))??-????1/??(??)??1/??(????)(1+??)/??(??).(3.50) By (3.42) and (3.50), we find ??????(??)+0?????0???????(??)=1+0?????0?????????(??)+??(??)(??)??(??)????+????0???/??0???????(??(??))??(??(??))????(??(??))????+??(??)??(??)-??1/??(??)??1/????(??)(1+??)/?????(??)+0?????0-?????(??)??(??)0?????0????1/??(??)??1/????(??)(1+??)/??(??).(3.51) In view of the above inequality, (3.5), (3.7), and ??(??)=??, we get ??????(??)+0?????0???????(??)=1+0?????0???(??)-??(??)??(??)2??-1??+??(??)??(??)-??1/??(??)??1/????(??)(1+??)/??+???(??)0?????0?????(??)??(??)-0?????0????1/??(??)??1/??(????)(1+??)/??(??),(3.52) which follows that ?????2???(??,??)??(??)??(??)2??-1-????1+0?????0?????(??)d??=-????2??(??,??)????(??)d??+????2-???(??,??)??(??)??(??)d??????2??(??,??)????(1+??)/??(??)??1/??(??)??1/?????(??)d??-0?????0?????2??(??,??)???+???(??)d??0?????0?????2?????(??,??)??(??)??(??)d??-0?????0?????2??(??,??)????(1+??)/??(??)??1/??(??)??1/??(??)d??.(3.53) Using the integration by parts formula and ??(??,??)=0, we have ?????2??(??,??)???(???)d??=-????,??2??????2?-?????2????(??,??)???????(??)d??,????2??(??,??)????(??)d??=-????,??2??????2?-?????2????(??,??)??????(??)d??.(3.54) So, by (3.53), we obtain ?????2???(??,??)??(??)??(??)2??-1-????1+0?????0?????(??)d??=????,??2??????2?+???0?????0?????,??2??????2?+?????2???(??,??)??(??)+????(??,??)/????????(??,??)??(??)d??-????2??(??,??)????(1+??)/??(??)??1/??(??)??1/??+???(??)d??0?????0?????2???(??,??)??(??)+????(??,??)/??????????(??,??)??(??)d??-0?????0?????2??(??,??)????(1+??)/??(??)??1/??(??)??1/??(??)d??.(3.55) Using the inequality ????-????(??+1)/??=????(??+1)??+1????+1????,(3.56) where ????=??1/??(??)??1/??(??),??=??(??)+????(??,??)/??????(??,??),(3.57) we have ?????2???(??,??)??(??)??(??)2??-1-????1+0?????0???(??(??)+??(??,??))d??=????,??2??????2?+???0?????0?????,??2??????2?(3.58) due to (3.55), which yields that 1?????,??2??????2???(??,??)??(??)??(??)2??-1-????1+0?????0?????(??(??)+??(??,??))d??=??2?+???0?????0?????2?,(3.59) which contradicts (3.34). The proof is complete.

From Theorem 3.7, we can obtain different oscillation conditions for all solutions of (1.1) with different choices of ??; the details are left to the reader.

Theorem 3.8. Assume that (1.6) and (3.30) hold. Furthermore, assume that 0=??(??)=??1<1. If ?8??0?1???(??)????0???(??)d??1/??d??=8,(3.60) then every solution ?? of (1.1) oscillates or lim???8??(??)=0.

Proof. Let ?? be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists ??1=??0 such that ??(??)>0,????(??(??))>0, and ??(??(??))>0 for all ??=??1. Then ??(??)>0 for ??=??1. Proceeding as in the proof of Theorem 3.1, we obtain (3.2). Thus ??(??)|??'(??)|??-1??'(??) is decreasing function, and there exists a ??2=??1 such that ??'(??)>0, ??=??2 or ??'(??)<0, ??=??2.
Case 1. Assume that ??'(??)>0, for ??=??2. Proceeding as in the proof of Theorem 3.3 and setting ??(??)=??, we can obtain a contradiction with (3.31).
Case 2. Assume that ??'(??)<0, for ??=??2. Then there exists a finite limit lim???8??(??)=??,(3.61) where ??=0. Next, we claim that ??=0. If not, then for any ??>0, we have ??<??(??)<??+??, eventually. Take 0<??<??(1-??1)/??1. We calculate ??(??)=??(??)-??(??)??(??(??))>??-??1??(??(??))>??-??1(??+??)=??(??+??)>????(??),(3.62) where ????=??+??-??1=???1-??1?-????1??+??>0.(3.63) From (3.2) and (3.62), we have ????(??)-????(??)????=??(??)????(??(??))=(????)????(??).(3.64) Integrating the above inequality from ??2 to ??, we get ???(??)-???(???)?????-??2??-??????2????=(????)???????2??(??)d??,(3.65) which implies ????1(??)=-???????(??)????2???(??)d??1/??.(3.66) Integrating the above inequality from ??2 to ??, we have ?????(??)=??2??-????????2?1???(??)????2???(??)d??1/??d??,(3.67) which yields ??(??)?-8; this is a contradiction. Hence, lim???8??(??)=0. Note that 0<??(??)=??(??). Then, lim???8??(??)=0. The proof is complete.

4. Examples

In this section, we will give two examples to illustrate the main results.

Example 4.1. Consider the following linear neutral equation: (??(??)+2??(??+(2??-1)??))??+??(??+(2??-1)??)=0,for??=??0,(4.1) where ?? and ?? are positive integers.
Let ??(??)=1,??(??)=2,??(??)=??+(2??-1)??,??(??)=1,??(??)=??+(2??-1)??.(4.2) Hence, ??(??)=1. Obviously, all the conditions of Corollary 3.5 hold. Thus by Corollary 3.5, every solution of (4.1) is oscillatory. It is easy to verify that ??(??)=sin?? is a solution of (4.1).

Example 4.2. Consider the following linear neutral equation: ?e2???1??(??)+2???(??+3)???+?e2??+1+12e2??-2???(??+1)=0,for??=??0,(4.3) where ?? and ?? are positive integers.
Let ??(??)=e2??1,??(??)=2,??(??)=e2??+1+e2??-2/2,??=1.(4.4) Clearly, all the conditions of Theorem 3.8 hold. Thus by Theorem 3.8, every solution of (4.3) is either oscillatory or lim???8??(??)=0. It is easy to verify that ??(??)=e-?? is a solution of (4.3).

Remark 4.3. Recent results cannot be applied to (4.1) and (4.3) since ??(??)=?? and ??(??)=??.

Remark 4.4. Using the method given in this paper, we can get other oscillation criteria for (1.1); the details are left to the reader.

Remark 4.5. It would be interesting to find another method to study (1.1) when ??°?????°??.

Acknowledgments

The authors sincerely thank the referees for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 60904024, 11026112), China Postdoctoral Science Foundation funded project (200902564), by Shandong Provincial Natural Science Foundation (ZR2010AL002, ZR2009AL003, Y2008A28), and also by University of Jinan Research Funds for Doctors (XBS0843).

References

  1. J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977. View at Zentralblatt MATH
  2. J. Baštinec, J. Diblík, and Z. Šmarda, “Oscillation of solutions of a linear second-order discrete-delayed equation,” Advances in Difference Equations, vol. 2010, Article ID 693867, 12 pages, 2010. View at Zentralblatt MATH
  3. J. Baštinec, L. Berezansky, J. Diblík, and Z. Šmarda, “On the critical case in oscillation for differential equations with a single delay and with several delays,” Abstract and Applied Analysis, vol. 2010, Article ID 417869, pp. 1–20, 2010.
  4. J. Diblík, Z. Svoboda, and Z. Šmarda, “Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case,” Computers & Mathematics with Applications. An International Journal, vol. 56, no. 2, pp. 556–564, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. R. P. Agarwal, S.-L. Shieh, and C.-C. Yeh, “Oscillation criteria for second-order retarded differential equations,” Mathematical and Computer Modelling, vol. 26, no. 4, pp. 1–11, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J.-L. Chern, W.-C. Lian, and C.-C. Yeh, “Oscillation criteria for second order half-linear differential equations with functional arguments,” Publicationes Mathematicae Debrecen, vol. 48, no. 3-4, pp. 209–216, 1996.
  7. J. Džurina and I. P. Stavroulakis, “Oscillation criteria for second-order delay differential equations,” Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 445–453, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. T. Kusano and N. Yoshida, “Nonoscillation theorems for a class of quasilinear differential equations of second order,” Journal of Mathematical Analysis and Applications, vol. 189, no. 1, pp. 115–127, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. T. Kusano and Y. Naito, “Oscillation and nonoscillation criteria for second order quasilinear differential equations,” Acta Mathematica Hungarica, vol. 76, no. 1-2, pp. 81–99, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. D. D. Mirzov, “The oscillation of the solutions of a certain system of differential equations,” Matematicheskie Zametki, vol. 23, no. 3, pp. 401–404, 1978.
  11. Y. G. Sun and F. W. Meng, “Note on the paper of Džurina and Stavroulakis,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 1634–1641, 2006. View at Publisher · View at Google Scholar
  12. B. Baculíková, “Oscillation criteria for second order nonlinear differential equations,” Archivum Mathematicum, vol. 42, no. 2, pp. 141–149, 2006. View at Zentralblatt MATH
  13. Ch. G. Philos, “Oscillation theorems for linear differential equations of second order,” Archiv der Mathematik, vol. 53, no. 5, pp. 482–492, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. Diblík, Z. Svoboda, and Z. Šmarda, “Retract principle for neutral functional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 12, pp. e1393–e1400, 2009. View at Publisher · View at Google Scholar
  15. L. Erbe, T. S. Hassan, and A. Peterson, “Oscillation of second order neutral delay differential equations,” Advances in Dynamical Systems and Applications, vol. 3, no. 1, pp. 53–71, 2008.
  16. L. Liu and Y. Bai, “New oscillation criteria for second-order nonlinear neutral delay differential equations,” Journal of Computational and Applied Mathematics, vol. 231, no. 2, pp. 657–663, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. R. Xu and F. Meng, “Some new oscillation criteria for second order quasi-linear neutral delay differential equations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 797–803, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. R. Xu and F. Meng, “Oscillation criteria for second order quasi-linear neutral delay differential equations,” Applied Mathematics and Computation, vol. 192, no. 1, pp. 216–222, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. J.-G. Dong, “Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3710–3717, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. B. Baculíková and D. Lacková, “Oscillation criteria for second order retarded differential equations,” Studies of the University of Žilina. Mathematical Series, vol. 20, no. 1, pp. 11–18, 2006.
  21. J. Jiang and X. Li, “Oscillation of second order nonlinear neutral differential equations,” Applied Mathematics and Computation, vol. 135, no. 2-3, pp. 531–540, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. Q. Wang, “Oscillation theorems for first-order nonlinear neutral functional differential equations,” Computers & Mathematics with Applications, vol. 39, no. 5-6, pp. 19–28, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. M. Hasanbulli and Y. V. Rogovchenko, “Oscillation criteria for second order nonlinear neutral differential equations,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4392–4399, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. S. H. Saker and D. O'Regan, “New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 423–434, 2011. View at Publisher · View at Google Scholar
  25. B. Baculíková and J. Džurina, “Oscillation of third-order neutral differential equations,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 215–226, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. Z. Han, T. Li, S. Sun, and Y. Sun, “Remarks on the paper, (Applied Mathematics and Computation), 2009, vol. 207, 388-396,” Applied Mathematics and Computation, vol. 215, pp. 3998–4007, 2010.
  27. Q. Zhang, J. Yan, and L. Gao, “Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 426–430, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. Z. Xu, “Oscillation theorems related to averaging technique for second order Emden-Fowler type neutral differential equations,” The Rocky Mountain Journal of Mathematics, vol. 38, no. 2, pp. 649–667, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. Z. Han, T. Li, S. Sun, and W. Chen, “On the oscillation of second-order neutral delay differential equations,” Advances in Difference Equations, vol. 2010, Article ID 289340, pp. 1–8, 2010. View at Zentralblatt MATH
  30. T. Li, Z. Han, P. Zhao, and S. Sun, “Oscillation of even-order neutral delay differential equations,” Advances in Difference Equations, vol. 2010, Article ID 184180, pp. 1–9, 2010.
  31. Z. Han, T. Li, S. Sun, C. Zhang, and B. Han, “Oscillation criteria for a class of second-order neutral delay dynamic equations of Emden-Fowler type,” Abstract and Applied Analysis, vol. 2011, Article ID 653689, pp. 1–26, 2011.
  32. Z. Han, T. Li, S. Sun, and C. Zhang, “An oscillation criteria for third order neutral delay differential equations,” Journal of Analytical and Applied, vol. 16, pp. 295–303, 2010. View at Publisher · View at Google Scholar
  33. S. Sun, T. Li, Z. Han, and Y. Sun, “Oscillation of second-order neutral functional differential equations with mixed nonlinearities,” Abstract and Applied Analysis, vol. 2011, pp. 1–15, 2011.
  34. Z. Han, T. Li, S. Sun, and W. Chen, “Oscillation criteria for second-order nonlinear neutral delay differential equations,” Advances in Difference Equations, vol. 2010, Article ID 763278, pp. 1–23, 2010. View at Zentralblatt MATH
  35. Z. Han, T. Li, C. Zhang, and Y. Sun, “Oscillation criteria for a certain second-order nonlinear neutral differential equations of mixed type,” Abstract and Applied Analysis, vol. 2011, pp. 1–8, 2011.