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

- Ercan Tunç
^{1}Email author

**2009**:896934

**DOI: **10.1155/2009/896934

© Ercan Tunç. 2009

**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

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.

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

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

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.

which is a contradiction.

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

which is a contradiction.

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.

on any interval where are nonoscillatory.

Proof.

which is a contradiction.

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

which is a contradiction.

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.

on any interval where , then is nonoscillatory.

Proof.

which is a contradiction.

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

which is a contradiction.

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

Theorem 3.2.

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

Proof.

which is a contradiction.

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

which is a contradiction.

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

Remark 3.3.

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.

which is a contradiction.

which is a contradiction.

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.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

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