Abstract and Applied Analysis

Volume 2014 (2014), Article ID 943170, 7 pages

http://dx.doi.org/10.1155/2014/943170

Research Article

New Oscillation Criteria for Third-Order Nonlinear Functional Differential Equations

Quanxin Zhang,¹ Li Gao,¹ Shouhua Liu,¹ and Yuanhong Yu²

¹Department of Mathematics, Binzhou University, Shandong 256603, China

²Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China

Received 19 November 2013; Accepted 13 January 2014; Published 11 March 2014

Academic Editor: Tongxing Li

Copyright © 2014 Quanxin Zhang 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.

This paper discusses oscillatory and asymptotic behavior of solutions of a class of third-order nonlinear functional differential equations. By using the generalized Riccati transformation and the integral averaging technique, three new sufficient conditions which insure that the solution is oscillatory or converges to zero are established. The results obtained essentially generalize and improve the earlier ones.

1. Introduction

As is well known, the comparison and separation theory of zeros distribution for second-order homogeneous linear differential equations established by Ladde et al. lays a foundation of oscillation theory for differential equations. During one and a half century, oscillation theory of differential equations has developed quickly and played an important role in qualitative theory of differential equations and the theory of boundary value problem. Oscillation theory of differential equations has been widely used in areas of physics, mechanics, radio technology, control system, sciences of life, economic relations, and population growth. The oscillations are physical phenomena which widely exist in physics and technological sciences, such as the oscillation of building and machine, electromagnetic vibration in radio technology and optical science, self-excited vibration in control system, sound vibration, beam vibration in synchrotron accelerator, the vibration sparked for burning rocket engine, and the complicated oscillation in chemical reaction. All different phenomena can be unified into an oscillation theory through an oscillation equation. There are many books on the oscillation theory, about which we can refer to [1].

The oscillation theory of third-order nonlinear functional differential equations has been widely applied in research of a lossless high-speed computer network and physical sciences. In this paper, we are concerned with oscillatory behavior of a third-order nonlinear functional differential equation as follows:

where is the ratio of positive odd integers. We have the following hypotheses:

 (A₁)  and satisfy

 (A₂) , such that , ; (A₃) , , and .

By a solution of (1), we mean a nontrivial function satisfying (1) which has the properties for and . Our attention is paid to those solutions of (1) which satisfy for all . A solution of (1) is said to be oscillatory on if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

In recent years, there have been numerous researches or many research activities concerning the oscillation and nonoscillation of solutions of three-order functional differential equations, which are special cases of (1), and for recent contributions, we refer to [2–8]. Consider

Parhi and Padhi [2] studied asymptotic behavior of solutions of (3). By using the integral averaging technique, Baculíková et al. [3] obtained sufficient conditions which insured that the solution of self-liner ordinary differential equation (4) was oscillatory or converges to zero. Mojsej [4] established the comparison results which insured that the solution of (5) was oscillatory or converges to zero. By the integral averaging technique, Saker [5] gave some oscillatory results of (5) when the condition holds. Several authors had proved some oscillatory results of (6) by method of comparison; see [6–8]. In this paper we intend to use Riccati transformation and the integral averaging technique to obtain some sufficient conditions which guarantee that every solution of (1) is oscillatory or converges to zero. Our results generalize and improve the corresponding theorems established in [3, 5].

2. Several Lemmas

Lemma 1. Assume that is a positive solution of (1). Then, there exists such that either

 (I) , , , ,

or

 (II) , , , .

The proof is similar to that of [3, Lemma 1] or [7, Lemma 1].

Throughout this paper, for sufficiently large , we denote

In order to make the definition of meaningful, we denote

Lemma 2. Assume that is a positive solution of (1) which satisfies case (I) in Lemma 1. Then there exists , such that

Assume that (8) and hold. Then

Proof. Pick so that for . Using (1), we obtain

Then, is strictly decreasing on . We get

and, hence, we have

By integrating both sides of the above inequality from to , it yields

Furthermore, by integrating both sides of (1) from to and noting that , , , we obtain

Then,

This completes the proof.

Lemma 3. Assume that is a positive solution of (1) which satisfies case (II) in Lemma 1. Furthermore,

Then, .

Proof. Assume that is a positive solution of (1) which satisfies case (II) in Lemma 1. Then, is decreasing and . We assert that . If not, then , . Integrating (1) from to , we get

Hence, we have

Integrating the above inequality from to , we obtain

Integrating the last inequality again from to , we have

Since condition (18) holds, we obtain , which contradicts . Hence, . This completes the proof.

3. Main Results

In this section, we obtain three new oscillatory criteria for (1) by using the generalized Riccati transformation and integral averaging technique of Philos-type [9]. Let

A function is said to have the property of and denote if it satisfies

(i), ; , ;(ii) and it is continuous.

The following are the main results of this paper.

Theorem 4. Let (8), (18), and hold. Assume that there exist , , and , such that

and for arbitrary , one has

where

Then, every solution of (1) is oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists a sufficiently large , such that , . By Lemma 1, we see that satisfies either case (I) or case (II).

If case (I) holds, then , . Define the function by

Then,

When holds, using (9) and (11), we get

In view of (28) and (29), noting that , we obtain

When holds, using (9) and (10), we have

In view of (28) and (31), which yields

From (30), (32), and the definition of , we get

By the definition of , we have

From (33) and (34), noting the definition of and , we obtain

Multiplying both sides of (35), with replaced by , by , integrating with respect to from to , we get

By integrating parts and using and (24), we obtain

Using averaging technique, we have

Combining (37) and (38), we get

which contradicts (25).

If case (II) holds, from (18), by Lemma 3, . This completes the proof.

Theorem 5. Let (8), (18), and hold. Assuming that there exist and, for all sufficiently large , there exists a , one has

where , , and are defined in Theorem 4. Then every solution of (1) is oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists a sufficiently large , such that , . By Lemma 1, we see that satisfies either case (I) or case (II).

If case (I) holds, we proceed in the proof of Theorem 4 and get (34). Then, from the definition of and , we obtain

By using the averaging technique, we find that

Hence, we get

Integrating (43) from to , we have

It follows that

which contradicts (40).

If case (II) holds, from (18), by Lemma 3, . This completes the proof.

By applying Theorem 5 with , , we have the following result.

Corollary 6. Let (8), (18), and hold, and for all sufficiently large , there exists a ; then, one has

where is defined in Theorem 4. Then every solution of (1) is oscillatory or converges to zero.

Theorem 7. Let (18) and hold. Assume that there exist , P, and , such that

and all sufficiently large such that

Then, every solution of (1) is oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that is a positive solution of (1). By Lemma 1, we see that satisfies either case (I) or case (II).

If case (I) holds, then , . Define the function by

Using (9), we have

Hence, by the definition of , we have

Multiplying both sides of (51), with replaced by , by and integrating with respect to from to , we get

Integrating by parts and using (47), which yields

Define and as follows:

where , , and . Using the inequality [10, Theorem 41]

we obtain

Combining (53) and (56), we get

which contradicts (48).

If case (II) holds, from (18), by Lemma 3, . This completes the proof.

Remark 8. If we let , in Theorem 4 and the function is of Theorem  3.3 in [5], then condition (25) is in [5]. Therefore, the result of in [5] is generalized to the case that is the ratio of positive odd integers. If we let in Theorem 7, the function is of Theorem  3.4 in [3], which condition (48) is converted to in [3]. Then, the result of in [3] is generalized to the one of (1) in this paper.

Example 9. Consider the three-order differential equation

where

Conditions (A₁), (A₂), and (A₃) are clearly satisfied. It is easy to find that (8) and (18) hold. Let , . Here

From Theorem 5, we have

so (40) is satisfied. Hence, by Theorem 5, every solution of (58) is oscillatory or converges to zero.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2013AM003), the Development Program in Science and Technology of Shandong Province of China (2010GWZ20401), and the Science Foundation of Binzhou University (BZXYKJ0810).

References

G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, vol. 110, Marcel Dekker, New York, NY, USA, 1987.

N. Parhi and S. Padhi, “Asymptotic behaviour of solutions of third order delay-differential equations,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 10, pp. 1609–1620, 2002.

B. Baculíková, E. M. Elabbasy, S. H. Saker, and J. Džurina, “Oscillation criteria for third-order nonlinear differential equations,” Mathematica Slovaca, vol. 58, no. 2, pp. 201–220, 2008.

I. Mojsej, “Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 11, pp. 3581–3591, 2008.

S. H. Saker, “Oscillation criteria of third-order nonlinear delay differential equations,” Mathematica Slovaca, vol. 56, no. 4, pp. 433–450, 2006.

B. Baculíková and J. Džurina, “Oscillation of third-order functional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 43, pp. 1–10, 2010.

B. Baculíková and J. Džurina, “Oscillation of third-order nonlinear differential equations,” Applied Mathematics Letters, vol. 24, no. 4, pp. 466–470, 2011.

S. R. Grace, R. P. Agarwal, R. Pavani, and E. Thandapani, “On the oscillation of certain third order nonlinear functional differential equations,” Applied Mathematics and Computation, vol. 202, no. 1, pp. 102–112, 2008.

C. G. Philos, “Oscillation theorems for linear differential equations of second order,” Archiv der Mathematik, vol. 53, no. 5, pp. 482–492, 1989.

G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1988.