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
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
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
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 
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 
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 
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 
Theorem 2.1. Assume that, for any
, there exist 
such that
for 
and (5)
holds. Let 
and nonnegative functions
satisfying
are continuous and
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
It is easy to verify that
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
According to Lemma 2.1, we obtain for 
Therefore, (21) yields
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 
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
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
then the results given in this paper are still valid.
3. Examples
Example 3.1. Consider
(23)
for
, where 
are constants. Let 
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
and
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
by Theorem 2.1.
Example 3.2. Consider the following forced quasilinear differential equation
(24)
for
, where 
are constants. Let 
and
, so 
The zeros of forcing term 
are
. For any 
we choose n sufficiently large so that 
, 
and 
Letting
, 
then we obtain
and
Therefore, Equation (14) is satisfied for i = 1 provided that
, where 
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.