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

Received: 27 July 2009

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)

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
(2.3)

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
(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)

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
(2.7)

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
(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)

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
(3.4)

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
(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)

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
(3.8)

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
(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)

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
(3.15)

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
(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

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Copyright

© Ercan Tunç. 2009

This article is published under license to BioMed Central Ltd. 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.