Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 545189, 13 pages

http://dx.doi.org/10.1155/2014/545189

Research Article

Set of Oscillation Criteria for Second Order Nonlinear Forced Differential Equations with Damping

Ambarka Abdalla Salhin, Ummul Khair Salma Din, Rokiah Rozita Ahmad, and Mohd Salmi Md Noorani

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 18 June 2014; Revised 13 August 2014; Accepted 19 August 2014; Published 2 September 2014

Academic Editor: Qi-Ru Wang

Copyright © 2014 Ambarka Abdalla Salhin 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.

By employing a generalized Riccati technique and an integral averaging technique, some new oscillation criteria are established for the second order nonlinear forced differential equation with damping. These results extend, improve, and unify some known oscillation criteria in the existing literature.

1. Introduction

The oscillatory problem for second order nonlinear forced differential equation with damping

is concerned, where and and H is a continuous function on .

Throughout this paper we will also suppose that there are positive constants , and satisfying the following:

), (), (), (), (A₅) for all ,() is continuous function such that and ,().

We will consider only nontrivial solutions of (1) which are defined for all large . A solution of (1) is said to be oscillatory if it has a sequence of zeros clustering at and nonoscillatory otherwise. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

In the late 19th century, some scholars focus on sufficient conditions for the oscillation theorems of different classes of differential equations with damping. We refer to the new published papers [1–11]. The oscillatory theory of second order nonlinear differential equations has been widely applied in research of lossless high-speed computer network and physical sciences.

Recently, the oscillatory behavior for various particular cases of (1), such as the nonlinear differential equations

has been studied extensively by numerous authors with different methods; see, for example, [9–11] and the references quoted therein.

In this paper, by using a generalized Riccati and integral averaging technique, several new oscillation criteria for (1) are established.

A significant drawback of many oscillation results for differential equations with damping reported in the literature is a necessity to impose a variety of additional restrictions on the sign of the damping term . We emphasize that our theorems are free of particular restrictions on .

2. Main Results

For convenience, we introduce the class of the function . Let . A function is said to belong to the class , if

(1) for and for ,(2) has continuous and nonpositive partial derivatives on with respect to the second variable,(3)there exists a function such that .

In this section, several oscillation conditions for (1) are established under the assumptions .

Theorem 1. Let assumptions be fulfilled and . If there exist functions and such that and

where

then (1) is oscillatory.

Proof. Let be a nonoscillatory solution of (1). Then there exists a such that for all . Without loss of generality, we may assume that on interval . A similar argument holds also for the case when is eventually negative. Defining a generalized Riccati transformation by

for all , then differentiating Equation (5), and using (1) and (A₁)–(A₆), it follows

for all with defined as above. Then we obtain

Multiplying both sides of (7) by , integrating it with respect to from to , and using the properties of the function , we get, for all ,

Therefore, for all ,

Applying inequality (9), for , yields

It follows that

which contradicts assumption (3), so (1) is oscillatory.

Corollary 2. If condition (3) is replaced by conditions

then (1) is oscillatory, where and are the same as defined in Theorem 1.

Example 3. Consider the nonlinear damped differential equation

where and . Since , , and the assumptions (A₁)–(A₆) hold. If we take and , then and . A direct computation yields that the conditions of Theorem 1 are satisfied; Example 3 is oscillatory.

Theorem 4. Let assumptions (A₁)–(A₆) be fulfilled and . Suppose that

If there exist functions and such that and

and for any

where and are the same as defined in Theorem 1, and

then (1) is oscillatory.

Proof. Let be a nonoscillatory solution of (1). Then there exists a such that for all . Without loss of generality, we may assume that on interval . A similar argument holds also for the case when is eventually negative.

Define the function as in (5). Similar to the proof of Theorem 1, we obtain inequality (9). Further, it follows

for and therefore

Thus, by (18), we get

for all . This implies that

Define

for all . Then

In order to show that

suppose that

By (15), there exists a positive constant such that

On the other hand, according to (27) for any positive constant there exists a such that

For ,

By (28) we can easily see that

Then there exists such that for all . Therefore, by (30), for all , and since is an arbitrary constant, we can make a conclusion that

Next, let us consider a sequence in with and such that

Now, by (25), there exists a constant such that

and hence (32) leads to

By taking into account (32), from (34), we derive

where . Thus

The above inequality and (35) imply that

Further, by Schwarz inequality, we have, for any positive integer ,

and therefore

It follows from (38) that

Consequently,

but the latter contradicts assumption (16). Hence, (27) fails to hold. Finally, by (23), we obtain

This contradicts the assumption (12). Therefore, (1) is oscillatory.

Theorem 5. Let assumptions (A₁)–(A₆) be fulfilled and . Suppose that (15) holds. If there exist functions and , such that , and (17) holds, and

and for every

where and are defined as in Theorem 1, , then (1) is oscillatory.

Proof. Without loss of generality, we assume that there exists a solution of (1) such that on for some . The function is defined as in (5). Then, following the proof of Theorem 4, we have (20). Now it follows that

for every . By (45), we know that (23) holds and

Then,

where and are defined as in the proof of Theorem 4 It follows from (44) and (45) that

Then there exists a sequence in such that and

Now, suppose that (27) holds. With the same argument as in Theorem 4, we conclude that (32) is satisfied. By (48), there exists a constant such that

Then, similar to the proof of Theorem 4, we obtain (41) which contradicts (50), and hence (27) fails. From (23) and (26) we have

which contradicts assumption (17).

Theorem 6. Let assumptions (A₁)–(A₆) be fulfilled and . Suppose that (15) holds. If there exist functions and , such that , and (17) and (45) hold, and

where and are defined as in Theorem 1 and , then (1) is oscillatory.

The proof of Theorem 6 is similar to the proof of Theorem 5.

Example 7. Consider the nonlinear damped differential equation

Obviously, for all and , is a constant. Since , , and , the assumptions (A₁)–(A₆) hold. If and , then and , for all .

A direct computation yields

We conclude by Theorem 6 that all solutions of this equation are oscillatory.

Remark 8. If (5) is replaced by

and (A₅) by for , we can obtain similar oscillation results that are derived in the present paper.

Remark 9. If we take , , it is easy to see that Theorems 1–6 reduce to Theorems 1−4 of Wang [8]. If we take , , and for , then (1) reduces to , and by taking , Theorems 1–5 reduce to Theorems 1–3 of Rogovchenko and Tuncay [6].

Remark 10. Theorems 1–6 and Corollary 2 are obtained by analogy with Theorems 1–4 from [9], and we do not require any restriction on the sign and differentiability of .

The following lemma will significantly simplify the proofs of next theorems. First recall class functions defined on . A function is said to belong to the class if

(i) for and when ,(ii) has partial derivatives on such that

for some .

Lemma 11. Let with , . If there exist and such that

then

for all , where

The proof of this lemma is similar to that of [4] and hence will be omitted.

In the next theorems we define the following functions that will be used in the proofs. Let

Theorem 12. Suppose that (A₁)–(A₇) hold. Assume that

If there exists a continuously differentiable function such that is nonnegative and decreasing function, we have

There exists an interval , and that there exists , and for any constant , such that

where

Then (1) is oscillatory.

Proof. Without loss of generality, we may assume that there exists a nonoscillatory solution of (1) such that on for some . The similar argument holds also for . Define the function as

Differentiating (70), using (1) and (A₁)–(A₇), we get

Integrating (71) from to we get that

Since , then by Bonnet’s Theorem, there exist for every such that

where is a constant. Then, we have, for ,

where .

Three cases of the oscillatory solutions are discussed below.

Case 1. Assume that is oscillatory; then there exists a sequence such that and , on . From (74) we get

Using (67) we obtain

Then there exists a constant , such that

Using Schwarz inequality, (A₇), and (77) we have

Applying (64),

where is a positive constant.

Let ; applying (78) we have

Using (78) in the above inequality leads to

Then, there exist constants and , such that

Substituting (82) in (71) we get

From (83) and by Lemma 11, we conclude that for any and

which contradicts condition (68).

Case 2. Assuming that for , then for , and by (74) we have

From (67) we see that

The following steps are similar to the proof of Case 1.

Case 3. Assume that for ; if (86) holds, then we have similar discussion in Case 2. If the integration in (86) is divergent, we can get the following inequality from (74) and (67):

where is a constant. By taking ,

From (87) and (88) we have

and from (86) we find

Integrating the above inequality, we obtain

Therefore

Applying (87) and (92), we get

Hence

The above inequality contradicts . This completes the proof.

Theorem 13. Suppose that (A₁)–(A₇) and (62)–(64) hold. Assume that there exists a function such that (65)–(68) and there exists such that

hold for all , where , and are the same as in Theorem 12. Then (1) is oscillatory.

Proof. Suppose that for all for some . Set in (23). We can get such that

Similarly, with (96) by setting , it follows that there exists such that

Clearly, we can observe that (68) is satisfied. Then (1) is oscillatory.

Example 14. Consider the following second order nonlinear differential equation with damping:

We can see that , , when . Let , and then

Moreover, and satisfy the conditions (65) and (66) of Theorem 12, respectively.

Remark 15. Theorems 12 and 13 are an extension of Theorems 3.1 and 3.2 of Lu and Meng [4] where they established several new oscillation criteria for (1) with , , , , and .

Theorem 16. Suppose that (A₁)–(A₆) hold and

Moreover, assume that there exists a continuous differentiable function such that for all and a constant . Then (1) is oscillatory if

Proof. On the contrary, we assume that (1) has a nonoscillatory solution . Without loss of the generality we suppose that for all . A similar argument holds also for the case when . Define the function as

This definition and (1) imply

From (A₁)–(A₆) and (101) we have

Then, for all ,

Note that

Substituting (107) in (102) we obtain

Then,

Dividing (109) by and taking the upper limit as , it contradicts with (102). This completes the proof.

Example 17. Consider the following second order nonlinear differential equation with damping:

We note that ,  . Considering and , then

Example 17 is oscillatory.

Remark 18. If , , , , , and , Theorem 16 reduces to Theorem 1 of Philos [5]. Furthermore, when , , and , then Theorem 16 reduces to Theorem 1 of Elabbasy and Elsharabasy [2].

Corollary 19. If condition (101) in Theorem 16 is replaced by

then Theorem 16 is still valid.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

This research has been completed with the support of these Grants: DIP-2012-31, FRGS/2/2013/SG04/UKM/02/3, and FRGS/1/2012/SG04/UKM/01/1.

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