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Discrete Dynamics in Nature and Society
Volume 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

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 [111]. 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, [911] 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 16 reduce to Theorems 1−4 of Wang [8]. If we take , , and for , then (1) reduces to , and by taking , Theorems 15 reduce to Theorems 1–3 of Rogovchenko and Tuncay [6].

Remark 10. Theorems 16 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):