Abstract and Applied AnalysisVolume 2011 (2011), Article ID 653689, 26 pagesdoi:10.1155/2011/653689

Research Article

Oscillation Criteria for a Class of Second-Order Neutral Delay Dynamic Equations of Emden-Fowler Type

Zhenlai Han,1,2 Tongxing Li,1,2 Shurong Sun,1,3 Chao Zhang,¹ and Bangxian Han⁴

¹School of Science, University of Jinan, Jinan, Shandong 250022, China²School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China³Department of Mathematics and Statistics, Missouri University of Science and Technology Rolla, Missouri 65409-0020, USA⁴Department of Mathematics, University of Science and Technology, Hefei 230026, China

Received 31 August 2010; Accepted 30 September 2010

Academic Editor: Elena Braverman

Copyright © 2011 Zhenlai Han et al. 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.

We establish some new oscillation criteria for the second-order neutral delay dynamic equations of Emden-Fowler type, on a time scale unbounded above. Here is a quotient of odd positive integers with a and p being real-valued positive functions defined on . Our results in this paper not only extend and improve the results in the literature but also correct an error in one of the references.

1. Introduction

The study of dynamic equations on time scales, which goes back to its founder Hilger [1], is an area of mathematics that has recently received a lot of attention. It was partly created in order to unify the study of differential and difference equations. Many results concerning differential equations are carried over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies and helps avoid proving results twice|once for differential equations and once again for difference equations.

The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [2]), that is, when and where Many other interesting time scales exist, and they give rise to many applications (see [3]). Dynamic equations on a time scale have an enormous potential for applications such as in population dynamics. For example, it can model insect populations that are continuous while in season, die out in, for example, winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population (see [3]). There are applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, and combinatorics. A recent cover story article in New Scientist [4] discusses several possible applications. Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [5] and references cited therein. A book on the subject of time scales, by Bohner and Peterson [3], summarizes and organizes much of time scale calculus; see also the book by Bohner and Peterson [6] for advances results of dynamic equations on time scales.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales unbounded above and neutral differential equations; we refer the reader to the papers [7–19]. Some authors are especially interested in obtaining sufficient conditions for the oscillation or nonoscillation of solutions of first and second-order linear and nonlinear neutral functional dynamic equations on time scales; we refer to the articles [20–28].

Agarwal et al. [7] considered the second-order delay dynamic equations and established some sufficient conditions for oscillation of (1.1). Şahiner [11] studied the second-order nonlinear delay dynamic equations and obtained some sufficient conditions for oscillation by employing Riccati transformation technique. Zhang and Zhu [13] examined the second-order dynamic equations and by using comparison theorems, they proved that oscillation of (1.3) is equivalent to the oscillation of the nonlinear dynamic equations and established some sufficient conditions for oscillation by applying the results established in [15]. Erbe et al. [16] investigated the oscillation of the second-order nonlinear delay dynamic equations and by employing the generalized Riccati technique, they established some new sufficient conditions which ensure that every solution of (1.5) oscillates or converges to zero. Mathsen et al. [20] investigated the first-order neutral delay dynamic equations and established some new oscillation criteria which as a special case involve some well-known oscillation results for first-order neutral delay differential equations. Zhu and Wang [21] studied the nonoscillatory solutions to neutral dynamic equations and gave a classification scheme for the eventually positive solutions of (1.7). Agarwal et al. [22], Şahíner [23], Saker et al. [24–26], Wu et al. [27], and Zhang and Wang [28] considered the second-order nonlinear neutral delay dynamic equations where is a quotient of odd positive integers, the delay function and satisfy and for all and and are real-valued positive functions defined on and and is continuous function such that for all and there exists a nonnegative function defined on such that

By employing different Riccati transformation technique, the authors established some oscillation criteria for all solutions of (1.8).

Recently, some authors have been interested in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of Emden-Fowler type dynamic equations on time scales, differential equations, and difference equations; see, for example, [29–47].

Han et al. [32] studied the second-order Emden-Fowler delay dynamic equations and established some sufficient conditions for oscillation of (1.9) and extended the results given in [7].

Saker [34] studied the second-order superlinear neutral delay dynamic equation of Emden-Fowler type on a time scale

The author assumes that the delay functions and satisfy for all and and are positive rd-continuous functions defined on such that and

The main result for the oscillation of (1.10) in [34] is the following.

Theorem 1.1 (see, [34, Theorem ]). Assume that - hold. Furthermore, assume that and there exists a -differentiable function such that for all constants Then every solution of (1.10) is oscillatory.

We note that in [34], the author gave an open problem, that is, how to establish oscillation criteria for (1.10) when

In [35], the author examined the oscillation of the second-order neutral delay dynamic equations

The author assumes that and as as and for each which are nondecreasing in and for where and with being odd integers.

The main result for the oscillation of (1.13) in [35] is the following.

Theorem 1.2 (see, [35, Theorem ]). Assume that ()–() hold. If for all sufficiently large then (1.13) oscillates.

We find that the conclusion of this theorem is wrong. The following is a counter example of this theorem.

Counter ExampleConsider the second-order differential equation Let For all sufficiently large we find that It is easy to see that Integrating by parts, we obtain Hence Therefore, by the above theorem, (1.15) is oscillatory. However, is a positive solution of (1.15). Therefore, the above theorem is wrong. Tracing the error to its source, we find that the following false assertion was used in the proof of the aforementioned theorem.

Assertion AIf is an eventually positive solution of (1.13), then is eventually positive.

Abdalla [37] studied the second-order superlinear neutral delay differential equations Most of the oscillation criteria are unsatisfactory since additional assumptions have to be imposed on the unknown solutions. Also, the author proved that if then every solution of (1.20) oscillates for every but one can easily see that this result cannot be applied when for

Lin [38] considered the second-order nonlinear neutral differential equations where The author investigated the oscillation for (1.22) when is superlinear.

Wong [46, 47] studied the second-order neutral differential equations whenever for all and are constants.

The main results for the oscillation of (1.23) in [46, 47] are the following.

Theorem 1.3 (see, [46, 47]). Suppose that is superlinear. Then a solution of (1.23) is either oscillatory or tends to zero if and only if

Theorem 1.4 (see, [46, 47]). Suppose that is sublinear and in addition satisfies Then a solution of (1.23) is either oscillatory or tends to zero if and only if

Li and Saker [40] investigated the second-order sublinear neutral delay difference equations where is a quotient of odd positive integers, for all and

The main result for the oscillation of (1.27) in [40] is the following.

Theorem 1.5 (see, [40, Theorem ]). Assume that there exists a positive sequence such that for every where Then every solution of (1.27) oscillates.

Yildiz and Öcalan [41] studied the higher-order sublinear neutral delay difference equations of the type where is a ratio of odd positive integers. The authors established some oscillation criteria of (1.29).

The main results for the oscillation of (1.29) when in [41] are the following.

Theorem 1.6 (see, [41, Theorem ]). Assume that and Then all solutions of (1.29) are oscillatory.

Theorem 1.7 (see, [41, Theorem ]). Assume that where is a constant, and Then every solution of (1.29) either oscillates or tends to zero as

Cheng [42] considered the oscillation of the second-order nonlinear neutral difference equations and established some oscillation criteria of (1.32) by means of Riccati transformation techniques.

Following this trend, in this paper, we are concerned with oscillation of the second-order neutral delay dynamic equations of Emden-Fowler type

As we are interested in oscillatory behavior, we assume throughout this paper that the given time scales are unbounded above; that is, it is a time scale interval of the form with

We assume that is a quotient of odd positive integers, the delay functions and satisfy for all and and are real-valued rd-continuous functions defined on

We note that if then and (1.33) becomes the second-order nonlinear delay differential equation

If then and (1.33) becomes the second-order nonlinear delay differential equation

In the case of (1.33) is the prototype of a wide class of nonlinear dynamic equations called Emden-Fowler sublinear dynamic equations, and if (1.33) is the prototype of dynamic equations called Emden-Fowler sublinear dynamic equations. It is interesting to study (1.33) because the continuous version, that is, (1.34), has several physical applications; see, for example, [1, 39], and when is a discrete variable, it is (1.35), and it is also important in applications.

2. Main Results

In this section, we give some new oscillation criteria of (1.33). In order to prove our main results, we will use the formula which is a simple consequence of Keller's chain rule [3, Theorem ]. Also, we need the following auxiliary results.

For the sake of convenience, we assume that

Lemma 2.1. Assume that (1.11) holds, and Then an eventually positive solution of (1.33) eventually satisfies that

Proof. From (1.11), the proof is similar to that of Saker et al. [24, Lemma ], so it is omitted.

Lemma 2.2. Assume that and Then an eventually positive solution of (1.33) eventually satisfies that or

Proof. Let be an eventually positive solution of (1.33). Then there exists such that and for all Assume that that is, Then, we have to show that (2.5) holds. It follows from (1.33) that which implies that is nonincreasing on Since the function is nondecreasing, must be nonincreasing on that is, is eventually either positive or negative. In both cases, is eventually monotonic, so that has a limit at infinity (finite or infinite). This implies that that is, is eventually positive (see [19, Lemma ]). Then we proceed as in the proof of [24, Lemma ] to obtain (2.5). The proof is complete.

Lemma 2.3. Assume that Further, is an eventually positive solution of (1.33). Then there exists a such that for

Proof. Let be an eventually positive solution of (1.33). Then there exists such that and for all It follows from (1.33) that (2.6) holds. From (2.6), we know that is an eventually decreasing function. We claim that eventually. Otherwise, if there exists a such that by (2.6), we have Thus Integrating the above inequality from to leads to which contradicts Hence, on Therefore, which yields Since is strictly decreasing, we have and so Also, we have that for large so we obtain Therefore, from (2.13), we have This completes the proof.

Lemma 2.4. Assume that Then an eventually positive solution of (1.33) satisfies that, for sufficiently large or

Proof. The proof is similar to that of the proof Lemmas 2.2 and 2.3, so we omit the details.

Theorem 2.5. Assume that (1.11) holds, and Then every solution of (1.33) oscillates if the inequality where has no eventually positive solution.

Proof. Suppose to the contrary that (1.33) has a nonoscillatory solution We may assume without loss of generality that there exists such that and for all From Lemma 2.1, there is some such that From (1.33), there exists a such that By Lemma 2.1, there exists a such that Substituting the last inequality in (2.21) we obtain for that Set Then from (2.23), is positive and satisfies the inequality (2.18), and this contradicts the assumption of our theorem. Thus every solution of (1.33) oscillates. This completes the proof.

By [41, Lemma ] and Theorem 2.5 in this paper, we have the following result.

Corollary 2.6. If is a positive integer, and then every solution of (1.33) oscillates if

Theorem 2.7. Assume that (2.4) holds, and and Then every solution of (1.33) either oscillates or tends to zero as if the inequality where has no eventually positive solution.

Proof. Suppose to the contrary that (1.33) has a nonoscillatory solution We may assume without loss of generality that there exists such that and for all From Lemma 2.2, if (i) holds, there is some such that From (1.33), there exists a such that By Lemma 2.2, there exists a such that Substituting the last inequality in (2.28), we obtain for that Set Then from (2.30), is positive and satisfies the inequality (2.25), and this contradicts the assumption of our theorem.If (ii) holds, by Lemma 2.2, we have This completes the proof.

By [41, Lemma ] and Theorem 2.7 in this paper, we have the following result.

Corollary 2.8. Assume that is a positive integer, and Then every solution of (1.33) either oscillates or tends to zero as if

Remark 2.9. Theorems 2.5 and 2.7 reduce the question of (1.33) to the absence of eventually positive solution (the oscillatory) of the differential inequalities (2.18) and (2.25).

Remark 2.10. From Theorem 2.5, Theorem 2.7, and the results given in [7–9, 12, 14], we can obtain some oscillation criteria for (1.33) in the case when

Theorem 2.11. Assume that (1.11) holds, and Then every solution of (1.33) oscillates if

Proof. We assume that (1.33) has a nonoscillatory solution such that and for all By proceeding as in the proof of Theorem 2.5, we get (2.21). By Lemma 2.1, note that and from Keller's chain rule, we obtain so Using (2.21), we have Hence, Upon integration we arrive at This contradicts (2.32) and finishes the proof.

Theorem 2.12. Assume that (2.4) holds, and and Then every solution of (1.33) either oscillates or tends to zero as if

Proof. By Lemma 2.2, the proof is similar to that of the proof of Theorem 2.11, so we omit the details.

Theorem 2.13. Assume that and Then every solution of (1.33) oscillates if holds for all sufficiently large

Proof. By Lemma 2.3, the proof is similar to that of the proof Theorem 2.11, so we omit the details.

Theorem 2.14. Assume that and Then every solution of (1.33) either oscillates or tends to zero as if holds for all sufficiently large

Proof. By using Lemma 2.4 and (2.28), the proof is similar to that of the proof of Theorem 2.11, so we omit the details.

Theorem 2.15. Assume that (1.11) holds, and Then every solution of (1.33) oscillates if

Proof. Suppose to the contrary that (1.33) has a nonoscillatory solution We may assume without loss of generality that there exists such that and for all By proceeding as in the proof of Theorem 2.5, we get (2.21). Thus from Lemma 2.1, we have for and hence This and Lemma 2.1 provide, for sufficiently large So We note that and imply This contradicts (2.41) and completes the proof.

Theorem 2.16. Assume that (2.4) holds, and and Then every solution of (1.33) either oscillates or tends to zero as if

Proof. By using Lemma 2.2 and (2.28), the proof is similar to that of the proof of Theorem 2.15, so we omit the details.

Theorem 2.17. Assume that Then every solution of (1.33) oscillates if holds for all sufficiently large

Proof. Suppose to the contrary that (1.33) has a nonoscillatory solution We may assume without loss of generality that there exists such that and for all By proceeding as in the proof of Theorem 2.5, we obtain (2.21). Thus from Lemma 2.3, we have, for and hence This and Lemma 2.3 provide, for sufficiently large So We note that and imply This contradicts (2.48) and completes the proof.

Theorem 2.18. Assume that (2.4) holds, and and Then every solution of (1.33) either oscillates or tends to zero as if holds for all sufficiently large

Proof. By using Lemma 2.4 and (2.28), the proof is similar to that of the proof of Theorem 2.17, so we omit the details.

Theorem 2.19. Assume that (1.11) holds, and Then every solution of (1.33) oscillates if

Proof. We assume that (1.33) has a nonoscillatory solution such that and for all By proceeding as in the proof of Theorem 2.5, we get (2.21). Define the function By Lemma 2.1, We calculate From (2.21), we have and by Lemma 2.1, we have because due to Keller's chain rule. Since thus Upon integration we arrive at Noting that we have This contradicts (2.55) and finishes the proof.

Theorem 2.20. Assume that (2.4) holds, and and Then every solution of (1.33) either oscillates or tends to zero as if

Proof. By using Lemma 2.2 and (2.28), the proof is similar to that of the proof of Theorem 2.19, so we omit the details.

In the following, we use a Riccati transformation technique to establish new oscillation criteria for (1.33).

Theorem 2.21. Assume that and Furthermore, suppose that there exists a positive -differentiable function such that for all sufficiently large and for all constants for Then every solution of (1.33) oscillates.

Proof. We assume that (1.33) has a nonoscillatory solution such that and for all By proceeding as in the proof of Theorem 2.5, we get (2.21). Define the function by the Riccati substitution Then By the product rule and then the quotient rule In view of (2.21) and (2.66), we have By the chain rule and we obtain where In view of we have and by Lemma 2.3, we see that Integrating (2.71) from to we obtain Hence which contradicts condition (2.65). The proof is complete.

Theorem 2.22. Assume that and If there exists a positive -differentiable function such that for all sufficiently large and for all constants for then every solution of (1.33) either oscillates or tends to zero as

Proof. By Lemma 2.4 and (2.28), the proof is similar to that of the proof of Theorem 2.21, so we omit the details.

Theorem 2.23. Assume that (1.11) holds, and Furthermore, suppose that there exists a positive -differentiable function such that for all sufficiently large and for all constants Then every solution of (1.33) oscillates.

Proof. We assume that (1.33) has a nonoscillatory solution such that and for all By proceeding as in the proof of Theorem 2.5, we obtain (2.21). Define the function by the Riccati substitution as (2.66). Then By the product rule and then the quotient rule In view of (2.21) and (2.66), we have From the chain rule and we get Noting that is nonincreasing, and there exists a constant such that hence we have In view of we have and by Lemma 2.1, we see that Integrating (2.81) from to we obtain Hence which contradicts condition (2.75). The proof is complete.

Theorem 2.24. Assume that (2.4) holds, If there exists a positive -differentiable function such that for all sufficiently large and for all constants then every solution of (1.33) either oscillates or tends to zero as

Proof. By Lemma 2.2 and (2.28), the proof is similar to that of the proof of Theorem 2.23, so we omit the details.

Theorem 2.25. Assume that and Furthermore, suppose that there exists a positive -differentiable function such that for all sufficiently large and for all constants for Then every solution of (1.33) oscillates.

Proof. We assume that (1.33) has a nonoscillatory solution such that and for all By proceeding as in the proof of Theorem 2.5, we have (2.21). Define the function by the Riccati substitution as (2.66). Then By the product rule and then the quotient rule In view of (2.21) and (2.66), we have By the chain rule and we obtain and noting that and there exists a constant such that so From there exists a positive constant such that Hence In view of we have and by Lemma 2.3, we see that Integrating (2.93) from to we obtain Thus which contradicts condition (2.85). The proof is complete.

Theorem 2.26. Assume that and If there exists a positive -differentiable function such that for all sufficiently large and for all constants for then every solution of (1.33) either oscillates or tends to zero as

Proof. By Lemma 2.4 and (2.28), the proof is similar to that of the proof of Theorem 2.25, so we omit the details.

3. Conclusions

In this paper, we consider the oscillation of second-order Emden-Fowler neutral delay dynamic equations (1.33). In some sense, our results extend and improve the results in [7, 32, 34, 35, 40, 41]. For example, Theorems 2.5, 2.11, 2.13, and 2.23 give some answers for the open problem posed by [34] since these results can be applied to (1.33) when Theorems 2.7, 2.12, 2.14, 2.16, 2.18, 2.20, 2.22, 2.24, and 2.26 correct an error in [35]. Theorem 2.15 includes the results of [7, Theorem ], [32, Theorem ], Theorem 2.11 includes the result of [32, Theorem ], Theorem 2.11 and Corollary 2.6 include the result of [41, Theorem ], Corollary 2.8 includes result of [41, Theorem ], Theorem 2.13 does not require the conditions so it improves the results of [40], and Theorems 2.17 and 2.21 improve the results in [34] since these results can be applied when

The main results in this paper require that it would be interesting to find another method to study (1.33) when Additional examples may also be given; due to the limited space, we leave this to the interested reader.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 60904024), China Postdoctoral Science Foundation funded project (20080441126, 200902564), Shandong Postdoctoral funded project (200802018) and supported by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), and by University of Jinan Research Funds for Doctors (XBS0843).

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