## Journal of Inequalities and Applications

Impact Factor 0.791

Open Access

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

Journal of Inequalities and Applications20092009:896934

DOI: 10.1155/2009/896934

Accepted: 6 November 2009

Published: 18 November 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
(1.1)
(1.2)

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, [137] as some related papers or books on the subject. In the most of these studies the following differential equation and some special cases of
(1.3)
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
(1.4)
and nonoscillation of all bounded solutions of the equations
(1.5)

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
(1.6)

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:
(1.7)

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
(2.1)
Integrating (2.1) from to , we obtain
(2.2)

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
(2.3)

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
(2.4)

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
(2.5)

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
(2.6)

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
(2.7)

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
(2.8)

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
(3.1)

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
(3.2)
Integrating (3.2) from to , we get
(3.3)

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
(3.4)

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
(3.5)

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
(3.6)

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
(3.7)

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
(3.8)

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
(3.9)

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:
(3.10)

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
(3.11)
But
(3.12)
Therefore
(3.13)
since for . So (3.11) yields
(3.14)

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
(3.15)

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
(3.16)
which is a contradiction. Let . Integrating (3.2) from to , we get
(3.17)

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:
(3.18)

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpaşa University

## References

1. 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)90166-0
2. Graef JR, Greguš M: Oscillatory properties of solutions of certain nonlinear third order differential equations. Nonlinear Studies 2000,7(1):43–50.
3. Hartman P: Ordinary Differential Equations, Classics in Applied Mathematics. SIAM, Philadelphia, Pa, USA; 2002.Google Scholar
4. Kartsatos AG, Manougian MN: Perturbations causing oscillations of functional-differential equations. Proceedings of the American Mathematical Society 1974, 43: 111–117. 10.1090/S0002-9939-1974-0328270-3
5. Nayak 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.
6. Padhi S: On oscillatory solutions of third order differential equations. Memoirs on Differential Equations and Mathematical Physics 2004, 31: 109–111.
7. Padhi S: On oscillatory linear third order forced differential equations. Differential Equations and Dynamical Systems 2005,13(3–4):343–358.
8. Parhi N: Nonoscillatory behaviour of solutions of nonhomogeneous third order differential equations. Applicable Analysis 1981,12(4):273–285. 10.1080/00036818108839368
9. Parhi N: Nonoscillation of solutions of a class of third order differential equations. Acta Mathematica Hungarica 1989,54(1–2):79–88. 10.1007/BF01950712
10. Parhi 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.
11. Parhi N: On non-homogeneous canonical third-order linear differential equations. Australian Mathematical Society Journal 1994,57(2):138–148. 10.1017/S1446788700037472
12. Parhi N, Das P: Oscillation criteria for a class of nonlinear differential equations of third order. Annales Polonici Mathematici 1992,57(3):219–229.
13. 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.
14. 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.
15. Parhi N, Das P: On nonoscillation of third order differential equations. Bulletin of the Institute of Mathematics Academia Sinica 1994,22(3):267–274.
16. Parhi N, Padhi S: On oscillatory linear differential equations of third order. Archivum Mathematicum, Universitatis Masarykianae Brunensis 2001,37(1):33–38.
17. 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.
18. 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.
19. 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.
20. Parhi N, Parhi S: On the behaviour of solutions of the differential equations . Polska Akademia Nauk. Annales Polonici Mathematici 1986,47(2):137–148.
21. 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.
22. 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.Google Scholar
23. 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.
24. Tunç C: On the nonoscillation of solutions of nonhomogeneous third order differential equations. Soochow Journal of Mathematics 1997,23(1):1–7.
25. 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.
26. 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.2006.02.011
27. Tunç C: Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations. Kuwait Journal of Science & Engineering 2005,32(1):39–48.
28. Tunç E: On the convergence of solutions of certain third-order differential equations. Discrete Dynamics in Nature and Society 2009, 2009:-12.Google Scholar
29. 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.
30. 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.
31. 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.
32. 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.
33. 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)60059-9
34. Zayed 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.
35. 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. 10.1137/0515084
36. Grace 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-F
37. Grace SR, Hamedani GG: On the oscillation of functional-differential equations. Mathematische Nachrichten 1999, 203: 111–123.