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Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 784279, 5 pages

http://dx.doi.org/10.1155/2013/784279

## New Interval Oscillation Criteria for Certain Linear Hamiltonian Systems

^{1}School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China^{2}Department of Mathematics, Jining University, Qufu 273155, China

Received 30 January 2013; Accepted 16 April 2013

Academic Editor: Wei Li

Copyright © 2013 Jing Shao and Fanwei Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Using a generalized Riccati transformation and the general integral means technique, some new interval oscillation criteria for the linear matrix Hamiltonian system , , are obtained. These results generalize and improve the oscillation criteria due to Zheng (2008). An example is given to dwell upon the importance of our results.

#### 1. Introduction

Hamiltonian matrix systems arise in many dynamic problems; the qualitative properties of Hamiltonian systems and matrix differential systems have been widely studied by many authors (e.g., see [1–17] and the references quoted there in). In this paper, we consider interval oscillatory properties for the linear Hamiltonian system: where , , and are matrices of real valued continuous functions on the interval, , , is positive definite, , and , are real valued continuous functions on . Here and in the sequel, the transpose of matrix is denoted by and its positive definiteness is denoted by . For any Hermitian matrix , its eigenvalues are real numbers, and we always denote them by . The trace of is denoted by , and recall that .

When , the system (1) reduces to the following linear self-adjoint matrix Hamiltonian system:

A solution of the system (1) (or (2)) is said to be nontrivial if for at least one . A nontrivial solution of the system (1) (or (2)) is said to be prepared or self-conjugate if , . System (1) (or (2)) is said to be oscillatory on if there is a nontrivial prepared solution of (1) such that vanishes at least once on for each . Otherwise, it is said to be nonoscillatory.

Firstly, we give some of the definitions for the sake of convenience.

*Definition 1. *Let be the linear space of real matrices, and let be the subspace of all symmetric matrices in . For any , means that is positive semidefinite and means that is positive definite.

*Definition 2. *A function is called to belong to a function class , denoted by , if , where ( is the closure of ) which satisfies , for . Furthermore, has continuous derivatives and on such that for all ,
where are functions determined by . Note that for ; and with , for , and belong to function class . For given , , a function is called to belong to a function class , denoted by , if , , for , and . We point out that if , and we define a function with for , where are constants, then .

*Definition 3. *A nonlinear (and possibly discontinuous) function with whenever is called negativity preserving; the class of all such negativity-preserving functionals on is denoted by .

The negativity-preserving functionals are widely used in oscillation criteria for matrix differential systems in paper [2]. Meanwhile, using negativity-preserving functionals, Zheng [11] considered the matrix system (2); oscillation criteria of interval type were obtained with the function as follows.

Theorem 4. *Suppose that there exists such that is differentiable, and . If there exists , such that
**
where and . Then system (2) is oscillatory.*

Theorem 5. *Suppose that . If, for each and for some ,
**
where , then system (2) is oscillatory.*

In this paper, using negativity-preserving functionals on a suitable matrix space and generalized Riccati transformation, we establish some new oscillation criteria for the system (1), which extend and improve the oscillation criteria mentioned previously, and an example is given which dwells upon the importance of our results.

#### 2. Main Results

Now, we give the main oscillation criteria for system (1) of interval type.

Theorem 6. *Suppose that there exists such that is differentiable, and . If there exists , such that
**
where
**
and is any constant nonzero symmetric matrix, then system (1) is oscillatory. *

*Proof. *Suppose that there exists a prepared solution of (1) which is not oscillatory. Without loss of generality, we may assume that for . Define
Then is Hermitian and satisfies the Riccati equation
Let . Then we get
It follows from (9) and (10) that, for ,
Since , we define . Let . Then
For each , by assumptions, there exist , , and satisfying . Now, multiplying (11) by and integrating it from to , we have
which implies that
Applying the functional to (14), we obtain a contradiction with assumption (6). This completes the proof.

Corollary 7. *Suppose that there exists such that is differentiable, and . If, for each , there exist , , and such that
**
for some , where is defined as previously mentioned, then system (1) is oscillatory.*

Corollary 8. *Suppose that there exists such that is differentiable, and . If there exists and, for each , there exist and such that
**
where is defined as that in Theorem 6, and
**
then system (1) is oscillatory. *

Let , ; we have the following useful corollary.

Corollary 9. *Suppose that there exists such that, is differentiable, and . If there exists such that for each ,
**
where is defined as in Theorem 6, and
**
then system (1) is oscillatory.*

Theorem 10. *Suppose that . If, for each and for some ,
**
where
**
and are defined as in Theorem 6, then system (1) is oscillatory.*

*Proof. *We choose , so , and
As in the proof of Theorem 6, we obtain (14), with being replaced by , replaced by , and the integral variable being replaced by , that is,
Suppose that , we get
So
As in the proof of [11, Theorem 2.7], we obtain
Dividing both sides by and taking the largest eigenvalue, we have
This contradicts with condition (20). The proof of Theorem 10 is completed.

#### 3. Examples

We give the following example to illustrate the applicability of our theorems.

*Example 1.* Consider the 4-dimensional system (1), where
We get that . Let ; let , and we have . For each , select to be large enough such that . Let . Using Theorem 6, we have
So we obtain that the Hamiltonian system is oscillatory for .

#### Acknowledgments

The authors thank the referees for giving some useful comments which improve their results. This research was partially supported by the NSF of China (Grants 11171178 and 11271225) and the Science and Technology Project of High Schools of Shandong Province (Grant J12LI52).

#### References

- S. G. Dubé and A. B. Mingarelli, “Note on a non-oscillation theorem of Atkinson,”
*Electronic Journal of Differential Equations*, vol. 2004, no. 22, pp. 1–6, 2004. View at Zentralblatt MATH · View at MathSciNet - A. B. Mingarelli, “Nonlinear functionals in oscillation theory of matrix differential systems,”
*Communications on Pure and Applied Analysis*, vol. 3, no. 1, pp. 75–84, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. G. A. Dubé and A. B. Mingarelli, “Nonlinear functionals and a theorem of Sun,”
*Journal of Mathematical Analysis and Applications*, vol. 308, no. 1, pp. 208–220, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. I. Al-Dosary, H. K. Abdullah, and D. Hussein, “Short note on oscillation of matrix Hamiltonian systems,”
*Yokohama Mathematical Journal*, vol. 50, no. 1-2, pp. 23–30, 2003. View at Zentralblatt MATH · View at MathSciNet - F. Meng and A. B. Mingarelli, “Oscillation of linear Hamiltonian systems,”
*Proceedings of the American Mathematical Society*, vol. 131, no. 3, pp. 897–904, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Yang and Y. Tang, “Oscillation theorems for self-adjoint matrix Hamiltonian systems involving general means,”
*Journal of Mathematical Analysis and Applications*, vol. 295, no. 2, pp. 355–377, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Meng and Y. Sun, “Interval criteria for oscillation of linear Hamiltonian systems,”
*Mathematical and Computer Modelling*, vol. 40, no. 7-8, pp. 735–743, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Yang, R. Mathsen, and S. Zhu, “Oscillation theorems for self-adjoint matrix Hamiltonian systems,”
*Journal of Differential Equations*, vol. 190, no. 1, pp. 306–329, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. G. Sun, “New oscillation criteria for linear matrix Hamiltonian systems,”
*Journal of Mathematical Analysis and Applications*, vol. 279, no. 2, pp. 651–658, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zheng, “Oscillation theorems for linear matrix Hamiltonian systems,”
*Applied Mathematics—A Journal of Chinese Universities*, vol. 17, pp. 285–290, 2002 (Chinese). - Z. Zheng, “Interval oscillation criteria for linear Hamiltonian systems,”
*Mathematische Nachrichten*, vol. 281, no. 11, pp. 1664–1671, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zheng, “Linear transformation and oscillation criteria for Hamiltonian systems,”
*Journal of Mathematical Analysis and Applications*, vol. 332, no. 1, pp. 236–245, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Meng and Y. Sun, “Oscillation of linear Hamiltonian systems,”
*Computers & Mathematics with Applications*, vol. 44, no. 10-11, pp. 1467–1477, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. H. Erbe, Q. Kong, and S. G. Ruan, “Kamenev type theorems for second-order matrix differential systems,”
*Proceedings of the American Mathematical Society*, vol. 117, no. 4, pp. 957–962, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Yang and Y. Tang, “Interval oscillation criteria for self-adjoint matrix Hamiltonian systems,”
*Proceedings of the Royal Society of Edinburgh. Section A*, vol. 135, no. 5, pp. 1085–1108, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. S. Kumari and S. Umamaheswaram, “Oscillation criteria for linear matrix Hamiltonian systems,”
*Journal of Differential Equations*, vol. 165, no. 1, pp. 174–198, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Kratz,
*Quadratic Functionals in Variational Analysis and Control Theory*, vol. 6 of*Mathematical Topics*, Akademie, Berlin, Germany, 1995. View at MathSciNet