Oscillation Theorems for a Class of Nonlinear Second Order Differential Equations with Damping ()
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).