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.
In this paper, we are concerned with the oscillatory behavior of solutions of the second-order nonlinear differential equations with damping
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
the nonlinear damped differential equation
and
have been studied extensively in recent years, see e.g. [1-21] and the references quoted therein. Moreover, in 2011, Wang [
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 [
Following Philos [
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
and for any,
where
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 [
for all, then differentiating Equation (2.6) and using Equation (1.1), we obtain
In view of (A1)-(A5), we get
for all with defined as above. Then we obtain
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,
Furthermore,
Now, it follows that
From (2.3) and (2.10), we have
for all and. Obviously,
for all (2.11)
and
Now, we can claim that
Otherwise,
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
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
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
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
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,
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 [
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.
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).