- Research Article
- Open Access
- Published:

# New Results on the Nonoscillation of Solutions of Some Nonlinear Differential Equations of Third Order

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 896934 (2009)

## Abstract

We give sufficient conditions so that all solutions of differential equations , and , are nonoscillatory. Depending on these criteria, some results which exist in the relevant literature are generalized. Furthermore, the conditions given for the functions and lead to studying more general differential equations.

## 1. Introduction

This paper is concerned with study of nonoscillation of solutions of third-order nonlinear differential equations of the form

where is a fixed real number, , , , , and such that and for all . are nondecreasing such that , for all , . Throughout the paper, it is assumed, for all and appeared in (1.1) and (1.2), that for all ; ; is a quotient of odd integers.

It is well known from relevant literature that there have been deep and thorough studies on the nonoscillatory behaviour of solutions of second- and third-order nonlinear differential equations in recent years. See, for instance, [1–37] as some related papers or books on the subject. In the most of these studies the following differential equation and some special cases of

have been investigated. However, much less work has been done for nonoscillation of all solutions of nonlinear functional differential equations. In this connection, Parhi [10] established some sufficient conditions for oscillation of all solutions of the second-order forced differential equation of the form

and nonoscillation of all bounded solutions of the equations

where the real-valued functions , , , , , and are continuous on with and ; , for ; , , and both and are quotients of odd integers.

Later, Nayak and Choudhury [5] considered the differential equation

and they gave certain sufficient conditions on the functions involved for all bounded solutions of the above equation to be nonoscillatory.

Recently, in 2007, Tunç [23] investigated nonoscillation of solutions of the third-order differential equations:

The motivation for the present work has come from the paper of Parhi [10], Tunç [23] and the papers mentioned above. We restrict our considerations to the real solutions of (1.1) and (1.2) which exist on the half-line , where () depends on the particular solution, and are nontrivial in any neighborhood of infinity. It is well known that a solution of (1.1) or (1.2) is said to be nonoscillatory on if there exists a such that for ; it is said to be oscillatory if for any there exist and satisfying such that and ; is said to be a -type solution if it has arbitrarily large zeros but is ultimately nonnegative or nonpositive.

## 2. Nonoscillation Behaviors of Solutions of (1.1)

In this section, we obtain sufficient conditions for the nonoscillation of solutions of (1.1).

Theorem 2.1.

Let . If , then all bounded solutions of (1.1) are nonoscillatory.

Proof.

Let be a bounded solution of (1.1) on , , such that for . Since , there exists a such that for . In view of the assumption , it follows that there exists a such that for . If possible, let be of nonnegative -type solution with consecutive double zeros at and () such that for . So, there exists such that and for . Multiplying (1.1) through by , we get

Integrating (2.1) from to , we obtain

which is a contradiction.

Let be of nonpositive -type solution with consecutive double zeros at and (). Then, there exists a such that and for .

Integrating (2.1) from to yields

which is a contradiction.

If possible, let be oscillatory with consecutive zeros at , and () such that , , , for and for . So there exists points and such that , , for and for . Now integrating (2.1) from to , we get

which is a contradiction. This completes the proof of Theorem 2.1.

Remark 2.2.

For the special case , , Theorem 2.1 has been proved by Tunç [23]. Our results include the results established in Tunç [23].

Theorem 2.3.

Let and , then all solutions of (1.1) which satisfy the inequality

on any interval where are nonoscillatory.

Proof.

Let be a solution of (1.1) on , . Due to , there exists a such that for . If possible, let be of nonnegative -type solution with consecutive double zeros at and () such that for . So, there exists a such that and for . Integrating (2.1) from to , we get

which is a contradiction.

Next, let be of nonpositive -type solution with consecutive double zeros at and (). Then, there exists such that and for .

Integrating (2.1) from to , we have

which is a contradiction.

Now, if possible let be oscillatory with consecutive zeros at , and () such that , , , for and for . Hence, there exist and such that and for and . Integrating (2.1) from to , we obtain

which is a contradiction. This completes the proof of Theorem 2.3.

Remark 2.4.

For the special case , , Theorem 2.3 has been proved by Tunç [25]. Our results include the results established in Tunç [25].

## 3. Nonoscillation Behaviors of Solutions (1.2)

In this section, we give sufficient conditions so that all solutions of (1.2) are nonoscillatory.

Theorem 3.1.

Suppose that and . If is a solution (1.2) such that it satisfies the inequality

on any interval where , then is nonoscillatory.

Proof.

Let be a solution of (1.2) on , . Due to , there exists a such that for . If possible, let be of nonnegative -type solution with consecutive double zeros at and () such that for . So, there exists a such that and for . Multiplying (1.2) through by , we get

Integrating (3.2) from to , we get

which is a contradiction.

Next, let be of nonpositive -type solution with consecutive double zeros at and (). Then, there exists such that and for .

Integrating (3.2) from to , we have

which is a contradiction.

Now, if possible let be oscillatory with consecutive zeros at , and () such that , , , for and for . Hence, there exist and such that and for and . Integrating (3.2) from to , we obtain

which is a contradiction. This completes the proof of Theorem 3.1.

Theorem 3.2.

Suppose that and on any subinterval of , . If is a solution of (1.2) such that it satisfies the inequality

on any subinteval of , , where , then is nonoscillatory.

Proof.

Let be a solution of (1.2) on , . Since , there exists a such that for . If possible, let be of nonnegative -type solution with consecutive double zeros at and () such that for . So, there exists a such that and for . Integrating (3.2) from to , we get

which is a contradiction.

Next, let be of nonpositive -type solution with consecutive double zeros at and (). Then, there exists such that and for .

Integrating (3.2) from to , we have

which is a contradiction.

Now, if possible let be oscillatory with consecutive zeros at , and () such that , , , for and for . Hence, there exist and such that and for and . Integrating (3.2) from to , we obtain

which is a contradiction. This completes the proof of Theorem 3.2.

Remark 3.3.

It is clear that Theorem 3.2 is not applicable to homogeneous equations:

where and .

Remark 3.4.

For the special case , , Theorem 3.2 has been proved by N. parhi and S. parhi [19, Theorem 2.7].

Theorem 3.5.

Let , and for all . If and are once continuously differentiable functions such that , and , then all solutions of (1.2) for which ultimately are nonoscillatory.

Proof.

Let be a solution of (1.2) on , , such that for . Since , there exists a such that for . If possible, let be of nonnegative -type solution with consecutive double zeros at and () such that for . So, there exists a such that and for . Integrating (3.2) from to , we get

But

Therefore

since for . So (3.11) yields

which is a contradiction.

Integrating (3.2) from to , we have

which is a contradiction.

Now, if possible let be oscillatory with consecutive zeros at , and () such that , , , for and for . So there exist and such that , and for . We consider two cases, namely, and . Suppose that . Integrating (3.2) from to , we get

which is a contradiction. Let . Integrating (3.2) from to , we get

We proceed as in nonnegative -type to conclude that . This is a contradiction. So is nonoscillatory. This completes the proof of Theorem 3.5.

Remark 3.6.

If in Theorem 3.5, then and hence the theorem is not applicable to homogeneous equation:

## References

Grace SR, Lalli BS:

**On oscillation and nonoscillation of general functional-differential equations.***Journal of Mathematical Analysis and Applications*1985,**109**(2):522–533. 10.1016/0022-247X(85)90166-0Graef JR, Greguš M:

**Oscillatory properties of solutions of certain nonlinear third order differential equations.***Nonlinear Studies*2000,**7**(1):43–50.Hartman P:

*Ordinary Differential Equations, Classics in Applied Mathematics*. SIAM, Philadelphia, Pa, USA; 2002.Kartsatos AG, Manougian MN:

**Perturbations causing oscillations of functional-differential equations.***Proceedings of the American Mathematical Society*1974,**43:**111–117. 10.1090/S0002-9939-1974-0328270-3Nayak PC, Choudhury R:

**Oscillation and nonoscillation theorems for third order functional-differential equation.***The Journal of the Indian Mathematical Society. (New Series)*1996,**62**(1–4):89–96.Padhi S:

**On oscillatory solutions of third order differential equations.***Memoirs on Differential Equations and Mathematical Physics*2004,**31:**109–111.Padhi S:

**On oscillatory linear third order forced differential equations.***Differential Equations and Dynamical Systems*2005,**13**(3–4):343–358.Parhi N:

**Nonoscillatory behaviour of solutions of nonhomogeneous third order differential equations.***Applicable Analysis*1981,**12**(4):273–285. 10.1080/00036818108839368Parhi N:

**Nonoscillation of solutions of a class of third order differential equations.***Acta Mathematica Hungarica*1989,**54**(1–2):79–88. 10.1007/BF01950712Parhi N:

**Sufficient conditions for oscillation and nonoscillation of solutions of a class of second order functional-differential equations.***Analysis*1993,**13**(1–2):19–28.Parhi N:

**On non-homogeneous canonical third-order linear differential equations.***Australian Mathematical Society Journal*1994,**57**(2):138–148. 10.1017/S1446788700037472Parhi N, Das P:

**Oscillation criteria for a class of nonlinear differential equations of third order.***Annales Polonici Mathematici*1992,**57**(3):219–229.Parhi N, Das P:

**On asymptotic property of solutions of linear homogeneous third order differential equations.***Unione Matematica Italiana. Bollettino B. Series VII*1993,**7**(4):775–786.Parhi N, Das P:

**Oscillatory and asymptotic behaviour of a class of nonlinear functional-differential equations of third order.***Bulletin of the Calcutta Mathematical Society*1994,**86**(3):253–266.Parhi N, Das P:

**On nonoscillation of third order differential equations.***Bulletin of the Institute of Mathematics Academia Sinica*1994,**22**(3):267–274.Parhi N, Padhi S:

**On oscillatory linear differential equations of third order.***Archivum Mathematicum, Universitatis Masarykianae Brunensis*2001,**37**(1):33–38.Parhi N, Padhi S:

**On oscillatory linear third order differential equations.***The Journal of the Indian Mathematical Society. (New Series)*2002,**69**(1–4):113–128.Parhi N, Parhi S:

**Oscillation and nonoscillation theorems for nonhomogeneous third order differential equations.***Bulletin of the Institute of Mathematics Academia Sinica*1983,**11**(2):125–139.Parhi N, Parhi S:

**Nonoscillation and asymptotic behaviour for forced nonlinear third order differential equations.***Bulletin of the Institute of Mathematics. Academia Sinica*1985,**13**(4):367–384.Parhi N, Parhi S:

**On the behaviour of solutions of the differential equations**.*Polska Akademia Nauk. Annales Polonici Mathematici*1986,**47**(2):137–148.Parhi N, Parhi S:

**Qualitative behaviour of solutions of forced nonlinear third order differential equations.***Rivista di Matematica della Università di Parma. Serie IV*1987,**13:**201–210.Swanson CA:

*Comparison and Oscillation Theory of Linear Differential Equations, Mathematics in Science and Engineering*.*Volume 48*. Academic Press, New York, NY, USA; 1968:viii+227.Tunç C:

**On the non-oscillation of solutions of some nonlinear differential equations of third order.***Nonlinear Dynamics and Systems Theory*2007,**7**(4):419–430.Tunç C:

**On the nonoscillation of solutions of nonhomogeneous third order differential equations.***Soochow Journal of Mathematics*1997,**23**(1):1–7.Tunç C:

**Non-oscillation criteria for a class of nonlinear differential equations of third order.***Bulletin of the Greek Mathematical Society*1997,**39:**131–137.Tunç C, Tunç E:

**On the asymptotic behavior of solutions of certain second-order differential equations.***Journal of the Franklin Institute, Engineering and Applied Mathematics*2007,**344**(5):391–398. 10.1016/j.jfranklin.2006.02.011Tunç C:

**Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations.***Kuwait Journal of Science & Engineering*2005,**32**(1):39–48.Tunç E:

**On the convergence of solutions of certain third-order differential equations.***Discrete Dynamics in Nature and Society*2009,**2009:**-12.Tunç E:

**Periodic solutions of a certain vector differential equation of sixth order.***The Arabian Journal for Science and Engineering A*2008,**33**(1):107–112.Tunç C:

**A new boundedness theorem for a class of second order differential equations.***The Arabian Journal for Science and Engineering A*2008,**33**(1):1–10.Zhong X-Z, Xing H-L, Shi Y, Liang J-C, Wang D-H:

**Existence of nonoscillatory solution of third order linear neutral delay difference equation with positive and negative coefficients.***Nonlinear Dynamics and Systems Theory*2005,**5**(2):201–214.Zhong X, Liang J, Shi Y, Wang D, Ge L:

**Existence of nonoscillatory solution of high-order nonlinear difference equation.***Nonlinear Dynamics and Systems Theory*2006,**6**(2):205–210.Zayed EME, El-Moneam MA:

**Some oscillation criteria for second order nonlinear functional ordinary differential equations.***Acta Mathematica Scientia B*2007,**27**(3):602–610. 10.1016/S0252-9602(07)60059-9Zayed EME, Grace SR, El-Metwally H, El-Moneam MA:

**The oscillatory behavior of second order nonlinear functional differential equations.***The Arabian Journal for Science and Engineering A*2006,**31**(1):23–30.Grace SR, Lalli BS, Yeh CC:

**Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term.***SIAM Journal on Mathematical Analysis*1984,**15**(6):1082–1093. 10.1137/0515084Grace SR:

**Oscillation criteria for forced functional-differential equations with deviating arguments.***Journal of Mathematical Analysis and Applications*1990,**145**(1):63–88. 10.1016/0022-247X(90)90432-FGrace SR, Hamedani GG:

**On the oscillation of functional-differential equations.***Mathematische Nachrichten*1999,**203:**111–123.

## Acknowledgment

The author would like to express sincere thanks to the anonymous referees for their invaluable corrections, comments, and suggestions.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Tunç, E. New Results on the Nonoscillation of Solutions of Some Nonlinear Differential Equations of Third Order.
*J Inequal Appl* **2009**, 896934 (2009). https://doi.org/10.1155/2009/896934

Received:

Accepted:

Published:

DOI: https://doi.org/10.1155/2009/896934

### Keywords

- Differential Equation
- Relevant Literature
- Nonlinear Differential Equation
- Homogeneous Equation
- Functional Differential Equation