New Oscillation Results for Forced Second Order Differential Equations with Mixed Nonlinearities ()
1. Introduction
The oscillatory behavior of second order differential equations has a major role in the theory of differential equations. It has been shown that many real world problems can be modelled, in particular, by half linear differential equations which can be regarded as a natural generalization of linear differential equations [1- 14]. A considerable amount of research has also been done on quasi-linear [15-18] and nonlinear second order differential equations [19-23].
In this paper, we investigate the oscillatory behavior of second order forced differential equation with mixed nonlinearities.
(1)
where
,
and
are real numbers,
and
might alternate signs.
By a solution of Equation (1), we mean a function
, where
depends on the particular solution, which has the property that
and satisfies Equation (1). We restrict our attention to the nontrivial solutions
of Equation (1) only, i.e., to solutions
such that
for all
. A nontrivial solution of (1) is oscillatory if it has arbitrarily large zeros, otherwise, it is called non-oscillatory. Equation (1) is said to be oscillatory if all its nontrivial solutions are oscillatory.
Equation (1) and its special cases such as the linear differential equation
(2)
the half-linear differential equation
(3)
and the quasi-linear differential equation
(4)
have been extensively studied by numerous authors with different methods (see, for example, [1-5,15-19] and the references quoted therein).
In 1999, Wong [1] proved the following theorem by making use of the “oscillatory intervals” of e(t) and Leighton’s variational principle (see [10]) for (2).
Theorem 1.1. Suppose that for any
, there exist
such that
(5)
Denote
![](https://www.scirp.org/html/6-7400658\a7d0cc99-3d28-4851-aed8-bdba3df6f231.jpg)
If there exist
such that
(6)
then Equation (2) is oscillatory.
Afterwards, in 2002, the authors of [2] extended Wong’s results, using a similar method, to Equation (3) as follows.
Theorem 1.2. Suppose that for any
, there exist
such that (5) holds. Let
![](https://www.scirp.org/html/6-7400658\d43e8a40-404e-47c8-ab07-1d3e946049b0.jpg)
If there exist
and a positive, nondecreasing function
such that
(7)
for i = 1, 2, where
, then (3) is oscillatory.
Later, in 2007, Zheng and Meng [16], considering a more general equation (4), improved the paper [2] and showed that the results obtained in [2] for Equation (3) can not be applied to the case
. The main result of Zheng and Meng [16] is the following.
Theorem 1.3. Assume that for any
, there exist
such that (5) holds. Let
![](https://www.scirp.org/html/6-7400658\6091c0d9-2c28-4d59-b41e-8472ef37eb00.jpg)
Suppose that there exist
and a positive, nondecreasing function
such that
(8)
for i =1, 2. Then Equation (4) is oscillatory, where
(9)
with the convention that ![](https://www.scirp.org/html/6-7400658\38fb90f9-1c89-4a1c-a2f7-ae1a75e6f477.jpg)
Also, in [2009], Zheng et al. [17] extended the results obtained for Equation (4) to Equation (1) as follows.
Theorem 1.4. Assume that for any
, there exist
such that
for
and (5) holds. Let
.
If there exist
and a positive function
such that
(10)
for i = 1, 2. Then Equation (1) is oscillatory, where
(11)
with the convention that ![](https://www.scirp.org/html/6-7400658\a12c0f51-16a2-450c-958e-8df2d00e8acb.jpg)
Recently, Shao [15] generalized the results of Zheng and Meng [16] by using the generalized variational principle due to Komkov [24] and gave the following result for Equation (4).
Theorem 1.5. Assume that, for any
, there exist
such that (1.5) holds. Let
, and nonnegative functions
satisfying
are continuous and ![](https://www.scirp.org/html/6-7400658\5dba1377-81ea-4f00-bbc4-bddb9b5171eb.jpg)
for
, i = 1, 2. If there exists a positive function
such that
(12)
for i = 1, 2, then Equation (4) is oscillatory, where
is the same as (9).
Motivated by the above theorems we propose some new oscillation results by employing the generalized variational principle and Riccati technique for Equation (1). Our results extend and generalize some known results in the literature. We now state our main results and several remarks.
2. New Oscillation Results
In order to prove our results we use the following wellknown inequality which is presented by Hardy et al. [25].
Lemma 2.1. (see [25]). If
and
are nonnegative, then
(13)
where equality holds if and only if ![](https://www.scirp.org/html/6-7400658\76d0522c-a864-4c4c-85f2-9a55562e108b.jpg)
Theorem 2.1. Assume that, for any
, there exist
such that
for
and (5)
holds. Let
and nonnegative functions
satisfying
![](https://www.scirp.org/html/6-7400658\5c0048ab-b626-430a-992b-0b954f2daaf8.jpg)
are continuous and
![](https://www.scirp.org/html/6-7400658\36f37c3c-4c83-4a58-8e6c-275d8c712810.jpg)
for
, i = 1, 2. If there exists a positive function
such that
(14)
for i =1, 2, then Equation (1) is oscillatory, where
is the same as (11).
Proof. Suppose that
is a nonoscillatory solution of Equation (1). Then, there exists a
such that
for all
. Without loss of generality, we may assume that
for all
. We introduce the Ricccati transformation
(15)
Differentiating (15) and using (1), we obtain, for all
,
(16)
By the assumption, we can choose
so that
on the interval
with
. As in [18], for given
, set
![](https://www.scirp.org/html/6-7400658\b8692a6a-6a26-4eef-b141-d7958a27c272.jpg)
It is easy to verify that
![](https://www.scirp.org/html/6-7400658\ec48eecf-04b4-4aab-8684-e7b561ddd32d.jpg)
So
obtains its minimum on
and
(17)
Then, by using (17) in (16), we get
(18)
Multiplying
through (18) and integrating over
, we have
(19)
By integration by parts and using the fact that
we have
(20)
In view of (19) and (20), we conclude that
(21)
Let
![](https://www.scirp.org/html/6-7400658\86d58a91-f47d-4683-9997-634b7ba7ddfd.jpg)
According to Lemma 2.1, we obtain for ![](https://www.scirp.org/html/6-7400658\885bde4c-4c74-4889-abe9-02f46103d0b1.jpg)
![](https://www.scirp.org/html/6-7400658\b06ca5fc-e55f-493e-a5ec-0634d0de3095.jpg)
Therefore, (21) yields
![](https://www.scirp.org/html/6-7400658\e6cd2ac2-97d6-4d22-b316-24a714f57582.jpg)
which contradicts the assumption (14) for
.
When
is a negative solution for
, we may employ the fact that
on
to reach a similar contradiction. Therefore, any solution
can be neither eventually positive nor eventually negative. Hence, any solution is oscillatory. This completes the proof of Theorem 2.1.
If
and
, then Equation (1) reduces to Equation (4). Thus by Theorem 2.1, we have the following oscillation result:
Corollary 2.1. Assume that, for any
, there exist
such that (5) holds. Let
, and nonnegative functions
satisfying
are continuous and ![](https://www.scirp.org/html/6-7400658\b0250f2f-92a3-4858-baab-f67e2b1d536b.jpg)
for
for i =1, 2. If there exists a positive function
such that
(22)
for i = 1, 2, then Equation (4) is oscillatory, where
is the same as (9).
Remark 1. Corollary 2.1 shows that Theorem 2.1 is a generalization of Theorem 1.5.
Remark 2. Let
in Corollary 2.1, then our main Theorem 2.1 reduces to Theorem 1.3.
Remark 3. If we choose
in Theorem 2.1, then we obtain Theorem 1.4.
Remark 4. If we choose
and
in Theorem 2.1, then we obtain Corollary 2.3 of Paper [17].
Remark 5. If we choose
and
in Corollary 2.1, then we obtain Corollary 2.3 of paper [16].
Remark 6. Let
![](https://www.scirp.org/html/6-7400658\5f7ca2b6-b37c-4f5f-a9ff-afd5770eefe7.jpg)
and
in Theorem 2.1, then Theorem 2.1 is a generalization of Theorem 1.1.
Remark 7. Let
If we choose
in Theorem 2.1, then Theorem 2.1 improves Theorem 1.2, since the positive constant
in Theorem 2.1 can be chosen as any number lying in
.
Remark 8. If the condition (5) in Theorem 2.1 and Corollary 2.1 is replaced by
![](https://www.scirp.org/html/6-7400658\a0675bff-189c-44cb-ad9b-d716b4b15d4b.jpg)
then the results given in this paper are still valid.
3. Examples
Example 3.1. Consider
(23)
for
, where
are constants. Let ![](https://www.scirp.org/html/6-7400658\563e4cbd-5333-452b-b032-fec19afd3bf6.jpg)
and
, so
. The zeros of forcing term
are
. For any
, we choose
sufficiently large so that
,
and
Letting
(it is easy to verify that
for
),
then we obtain
![](https://www.scirp.org/html/6-7400658\680cacd4-b34e-4f61-978b-e656197f6371.jpg)
and
![](https://www.scirp.org/html/6-7400658\00ba8de6-45ea-42f4-8d85-99d5fe372f1d.jpg)
Therefore, Equation (14) is satisfied for i = 1 provided that
In a similar way, for
and
, we choose
,
(it is easy to verify that
for
) so that that (14) is valid for i = 2. Thus (23) is oscillatory for
![](https://www.scirp.org/html/6-7400658\d5b34c20-27b3-4bce-a14f-5a45d1238c6b.jpg)
by Theorem 2.1.
Example 3.2. Consider the following forced quasilinear differential equation
(24)
for
, where
are constants. Let ![](https://www.scirp.org/html/6-7400658\03e1017b-392b-4e7c-b9a7-7e72a882bd68.jpg)
and
, so
The zeros of forcing term
are
. For any ![](https://www.scirp.org/html/6-7400658\27b9e15a-c2dd-47f6-8e3a-7e23f3ffb387.jpg)
we choose n sufficiently large so that
,
and
Letting
![](https://www.scirp.org/html/6-7400658\29b5067c-df26-47a5-90e9-e4eb58d00fda.jpg)
,
then we obtain
![](https://www.scirp.org/html/6-7400658\e4715348-0974-4369-bc62-351f6672e93f.jpg)
and
![](https://www.scirp.org/html/6-7400658\858dd100-754b-4696-b9c3-0b87003542f3.jpg)
Therefore, Equation (14) is satisfied for i = 1 provided that
, where ![](https://www.scirp.org/html/6-7400658\bd968847-3fbc-4795-95a9-71f23b10c960.jpg)
In a similar way, for
and
, we choose
,
so that (14) is valid for i = 2. Thus (24) is oscillatory for
by Theorem 2.1.
4. Conclusion
The oscillatory behavior of many different kinds of differential equations has been investigated and a great deal of results has been obtained in the literature. In this article, we generalized the results obtained in [16,17] and extended the results of Shao [15] by using the generalized variational principle and Riccati tecnique. In a similar way, the results obtained for Equation (1) can be extended to a more general class of differential equations.
5. Acknowledgements
The authors would like to express sincere thanks to the anonymous referee for her/his invauable corrections, comments and suggestions on the paper.