ABSTRACT

Some new oscillation criteria are given for forced second order differential equations with mixed nonlinearities by using the generalized variational principle and Riccati technique. Our results generalize and extend some known oscillation results in the literature.

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.

REFERENCES