Abstract

The purpose of this paper is to develop a generalized matrix Riccati technique for the selfadjoint matrix Hamiltonian system , . By using the standard integral averaging technique and positive functionals, new oscillation and interval oscillation criteria are established for the system. These criteria extend and improve some results that have been required before. An interesting example is included to illustrate the importance of our results.

1. Introduction

In this paper, we consider oscillatory properties for the linear Hamiltonian system where , and are real matrix-valued functions, are Hermitian, and is positive definite. By , we mean the conjugate transpose of the matrix , for any Hermitian matrix .

For any two solutions and of system (1.1), the Wronski matrix is a constant matrix. In particular, for any solution of system (1.1), is a constant matrix.

A solution of system (1.1) is said to be nontrivial if is fulfilled for at least one . A nontrivial solution of system (1.1) is said to be conjoined (prepared) if . A conjoined solution of system (1.1) is said to be a conjoined basis of system (1.1) if the rank of the matrix is .

In 2000, Kumari and Umamaheswaram [1], Yang and Cheng [2], and Wang [3] used the substitution to study the oscillation of system (1.1). One of the main results in [1] is as follows.

Theorem A. Let and . Let the functions and satisfy the following three conditions: (i), for on ; (ii) has a continuous and nonpositive partial derivative on with respect to the second variable; (iii), for all .
If there exists a function such that where is the identity matrix, and then, system (1.1) is oscillatory.

In 2003, Meng and Mingarelli [4], Wang [3], and Zheng and Zhu [5] studied the oscillation of system (1.1) by using the substitution One of the main results in [4] is as follows.

Theorem B. Let the functions and satisfy (i)–(iii) in Theorem A and, for all sufficiently large . Assume that there exist a function and a monotone subhomogeneous functional of degree on such that where , is a fundamental matrix of the linear equation , and Then, system (1.1) is oscillatory.

In 2004, Sun and Meng [6] also studied the oscillation of system (1.1). One of the main results in [6] is as follows.

Theorem C. Let be as in Theorem A, and suppose that If there exist a function and a positive linear functional on such that and suppose also that there exists a function such that for all and where and are the same as in Theorem A, then, the system (1.1) is oscillatory.

Recently, Li et al. [7] also studied the oscillation of system (1.1) by using the standard integral averaging technique and the substitution where is as in (1.5). One of the main results in [7] is as follows.

Theorem D. Let be as in Theorem A, and suppose that there exist a function and a positive linear functional on , for some , such that where Then, system (1.1) is oscillatory.

The purpose of this paper is further to improve Theorems A, B, C, and D as well as other related results regarding the oscillation of the system (1.1), by refining the standard integral averaging technique and Riccati transformation.

Now we use the general weighted functions from the class . Let and . We say that a continuous function belongs to the class if (i) for , on , (ii) has a continuous and nonpositive partial derivative on with respect to the second variable, (iii), for all , where .

We now follow [8] in defining the space as the real linear spare of all real symmetric matrices. Let be a linear functional on , is said to be positive if whenever and .

2. Main Results

In this paper, we need the following lemma.

Lemma 2.1 (see [6]). If is a positive linear functional on , then, for all , one has

Theorem 2.2. Let . If there exist a function , a matrix function , and a positive linear functional on , for some , such that where , and , then, system (1.1) is oscillatory.

Proof. Assume to the contrary that system (1.1) is nonoscillatory. Then, there exists a nontrivial prepared solution of such that is nonsingular for all sufficiently large . Without loss of generality, we assume that for all . This allows us to make a Riccati transformation for all . Then, is well defined, Hermitian, and solves the Riccati equation on .
Let . So, from (2.4), we have Now by the substitution in (2.5), we obtain By rearranging the terms, we get Multiplying (2.7), with replaced by , by and integrating from and , we obtain Taking the linear functional on both sides of the above equation, we have, for some , So, Taking the upper limit in both sides of (2.10) as , we obtain which contradicts (2.2). This completes the proof of Theorem 2.2.

Theorem 2.3. Let the functions and be as in Theorem 2.2, and suppose that If there exists a function , such that, for all , and for some , where , and are the same as in Theorem 2.2, then, system (1.1) is oscillatory.

Proof. Assume to the contrary that system (1.1) is nonoscillatory. Similar to the proof of Theorem 2.2, we can obtain, for all , and for some , Taking the upper limit of the above inequation as , By (2.13), we obtain Besides, we have Now, we claim that Suppose to the contrary that By (2.12), there exists a positive constant satisfying And according to the above , there exists such that Thus, From (2.22), there exists a such that, for all , So, Since is arbitrary, we get which contradicts (2.19). So, (2.20) holds; then, by (2.18) and (2.20), we can obtain which contradicts (2.14). This completes our proof of Theorem 2.3.

Example 2.4. Consider the linear Hamiltonian system (1.1), where are -matrices and are Hermitian.
Let , and , where is a -matrix. Then, , , , , and . According to Theorem 2.3, we get that this linear system is oscillatory.

Remark 2.5. In Theorem 2.2, let , . Theorem 2.2 reduces to Theorem D. In Theorem 2.3, we obtain the same result in which we remove the two assumptions (1.9) in Theorem C. Therefore, Theorems 2.2 and 2.3 are generalizations and improvements of [7, Theorem 2.1] and [6, Theorem 3].

Remark 2.6. The above theorems give rather wide possibilities of deriving different explicit oscillation criteria for system (1.1) with appropriate choices of the functions , and . For example, we can obtain some useful oscillation criteria if we choose , or , and so forth.

3. Interval Oscillation Criteria

Now we establish interval oscillation criteria of system (1.1), that is, criteria given by the behavior of system (1.1) only on a sequence of subinterval of . We assume that a function satisfying (i). Further, we assume that and has partial derivatives and on such that where .

We first prove two lemmas.

Lemma 3.1. Suppose that is a nontrivial prepared solution of system (1.1) such that on . Then, for any , matrix function , satisfies (i), (3.1) and (3.2), and a positive linear functional on , one has, for some , where is defined by (2.3) on , , and are the same as in Theorem 2.2.

Proof. Since is a nontrivial prepared solution of system (1.1) such that is nonsingular on , then, by (2.3) is well defined and solves the Riccati equation (2.7) on .
On multiplying (2.7) by and integrating with respect to from to for , we can find Taking the linear functional on both sides of the above equation, we have, for some , That is, Let ,