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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 784279, 5 pages
New Interval Oscillation Criteria for Certain Linear Hamiltonian Systems
1School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
2Department 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.
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.
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 . Meanwhile, using negativity-preserving functionals, Zheng  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.
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.
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 .
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).
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