ABSTRACT

The oscillatory behavior of solutions of a class of second order nonlinear differential equations with damping is studied and some new sufficient conditions are obtained by using the refined integral averaging technique. Some well known results in the literature are extended. Moreover, two examples are given to illustrate the theoretical analysis.

1. Introduction

In this paper, we are concerned with the oscillatory behavior of solutions of the second-order nonlinear differential equations with damping

(1.1)

where and .

In what follows with respect to Equation (1.1), we shall assume that there are positive constants and satisfying

(A1) and for all;

(A2) for all;

(A3) and for all;

(A4) and;

(A5) for all.

We shall consider only nontrivial solutions of Equation (1.1) which are defined for all large t. A solution of Equation (1.1) is said to be oscillatory if it has arbitrarily large zeros, otherwise it is said to be nonoscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory.

The oscillation problem for various particular cases of Equation (1.1) such as the nonlinear differential equation

, (1.2)

the nonlinear damped differential equation

(1.3)

and

(1.4)

have been studied extensively in recent years, see e.g. [1-21] and the references quoted therein. Moreover, in 2011, Wang [22] established some oscillation criteria for Equation (1.1) firstly, some new sharper results are obtained in the present paper.

An important method in the study of oscillatory behaviour for Equations (1.1)-(1.4) is the averaging technique which comes from the classical results of Wintner [19] and Hartman [18]. Using the generalized Riccati technique and the refined integral averaging technique introduced by Rogovchenko and Tuncay [20,21], several new oscillation criteria for Equation (1.1) are established in Section 2, we also show some examples to explain the application of our oscillation theorems in Section 2. Our results strengthen and improve the recent results of [1] and [21,22].

2. The Main Results

Following Philos [10], let us introduce now the class of functions which will be extensively used in the sequel. Let

and.

The function is said to belong to the class if 1) for; on;

2) has a continuous and nonpositive partial derivative on with respect to the second variable;

3) There exists a function such that

.

In this section, several oscillation criteria for Equation (1.1) are established under the assumptions (A1)-(A5). The first result is the following theorem.

Theorem 2.1. Let assumption (A1)-(A5) be fulfilled and. If there exists functions

such that and

(2.1)

(2.2)

and for any,

(2.3)

where

(2.4)

(2.5)

and, then Equation (1.1) is oscillatory.

Proof. Let be a nonoscillatory solution of Equation (1.1). 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. As in [1], define a generalized Riccati transformation by

(2.6)

for all, then differentiating Equation (2.6) and using Equation (1.1), we obtain

(2.7)

In view of (A1)-(A5), we get

for all with defined as above. Then we obtain

. (2.8)

On multiplying Equation (2.8) (with t replaced by s) by, integrating with respect to s from T to t for, using integration by parts and property 3), we get

Then, for any

and, for all,

(2.9)

Furthermore,

Now, it follows that

(2.10)

From (2.3) and (2.10), we have

for all and. Obviously,

for all         (2.11)

and

(2.12)

Now, we can claim that

, (2.13)

Otherwise,

. (2.14)

By (2.1), there exists a positive constant such that

and there exists a satisfying

for all.

On the other hand, by (2.14) for any, there exists a such that

for all.

Using integration by parts, we obtain

This implies that

for all.

Since is an arbitrary positive constant, we get

which contradicts (2.12), so (2.13) holds, and from (2.11)

which contradicts (2.2), then Equation (1.1) is oscillatory.

Now, we define, here . Evidently, and

.

Thus, by Theorem 2.1, we obtain the following result.

Corollary 2.1. Let assumption (A1)-(A5) be fulfilled. Suppose that (2.2) holds. If there exist functions such that,

where and are defined as in Theorem 2.1, then Equation (1.1) is oscillatory.

Example 2.1. Consider the nonlinear damped differential equation

.

where and, , ,

.

The assumptions (A1)-(A5) hold. If we take, and, then, and

.

A direct computation yields that the conditions of Corallary 2.1 are satisfied, Equation (1.1) is oscillatory.

As a direct consequence of Theorem 2.1, we get the following result.

Corollary 2.2. In Theorem 2.1, if condition (2.3) is replaced by

where and are the same as in Theorem 2.1, then Equation (1.1) is oscillatory.

Theorem 2.2. Let assumption (A1)-(A5) be fulfilled. For some, if there exist functions

such that

and

where is the same as in Theorem 2.1, and

(2.16)

Then Equation (1.1) is oscillatory.

Proof. Let be a nonoscillatory solution of Equation (1.1). 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 (2.6). Using (A1)-(A5) and (2.7), we have

(2.17)

where is the same as in Theorem 2.1. On the other hand, since the inequality

holds for all and. Let

we get from (2.17) that

(2.18)

On multiplying (2.18) (with t replaced by s) by, integrating with respect to s from T to t for and, using integration by parts and property 3), we get

This implies that

Using the properties of, we have

Therefore,

for all, and so

which contradicts with the assumption (2.15). This completes the proof of Theorem 2.2.

Let, from Theorem 2.2, we obtain the next result.

Corollary 2.3. Let assumption (A1)-(A5) be fulfilled. If there exist functions such that

and

holds for some integer and, where and are defined as in Theorem 2.2, then Equation (1.1) is oscillatory.

Example 2.2. Consider the nonlinear damped differential equation

(2.19)

Evidently, for all, and, we have

and

.

Let, then

and.

Therefore, Equation (2.19) is oscillatory by Corallary 2.3.

Theorem 2.3. Let assumption (A1)-(A5) be fulfilled and. If there exist functions

such that (2.1) holds and, and for all, any, and for some,

(2.20)

where and are the same as in Theorem 2.2 and. If (2.2) is satisfied, then Equation (1.1) is oscillatory.

Proof. The proof of this theorem is similar to that of Theorem 2.1 and hence is omitted.

Theorem 2.4. Let all assumptions of Theorem 2.3 be fulfilled except the condition (2.20) be replaced by

then Equation (1.1) is oscillatory.

Remark 2.1. If we take, then the condition is not necessary.

Remark 2.2. If we take, then Theorem 2.3 and 2.4 reduce to Theorem 9 and 10 of [21] with, respectively.

Remark 2.3. If replace (A5) and (2.6) by exists, for and define

respectively, we can obtain similar oscillation results that are derived in the present paper.

3. Acknowledgements

This work was supported by the National Natural Science Foundation of China (11071011), the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR201107123), the Plan Project of Science and Technology of Beijing Municipal Education Committee (KM201210016007) and the Natural Science Foundation of Beijing University of Civil Engineering and Architecture (10121907).

REFERENCES