Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 942042 | 13 pages | https://doi.org/10.1155/2012/942042

Some Oscillation Results for Linear Hamiltonian Systems

Academic Editor: Mehmet Sezer
Received08 Sep 2011
Accepted07 Dec 2011
Published19 Feb 2012

Abstract

The purpose of this paper is to develop a generalized matrix Riccati technique for the selfadjoint matrix Hamiltonian system 𝑈=𝐴(𝑡)𝑈+𝐵(𝑡)𝑉, 𝑉=𝐶(𝑡)𝑈𝐴(𝑡)𝑉. 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 system𝑈𝑉=𝐴(𝑡)𝑈+𝐵(𝑡)𝑉,=𝐶(𝑡)𝑈𝐴(𝑡)𝑉,𝑡𝑡0,(1.1) where 𝐴(𝑡),𝐵(𝑡), and 𝐶(𝑡) are real 𝑛×𝑛 matrix-valued functions, 𝐵,𝐶 are Hermitian, and 𝐵 is positive definite. By 𝑀, we mean the conjugate transpose of the matrix 𝑀, for any 𝑛×𝑛 Hermitian matrix 𝑀.

For any two solutions (𝑈1(𝑡),𝑉1(𝑡)) and (𝑈2(𝑡),𝑉2(𝑡)) of system (1.1), the Wronski matrix 𝑈1(𝑡)𝑉2(𝑡)𝑉1(𝑡)𝑈2(𝑡) is a constant matrix. In particular, for any solution (𝑈(𝑡),𝑉(𝑡)) of system (1.1), 𝑈(𝑡)V(𝑡)𝑉(𝑡)𝑈(𝑡) is a constant matrix.

A solution (𝑈(𝑡),𝑉(𝑡)) of system (1.1) is said to be nontrivial if det𝑈(𝑡)0 is fulfilled for at least one 𝑡𝑡0. A nontrivial solution (𝑈(𝑡),𝑉(𝑡)) of system (1.1) is said to be conjoined (prepared) if 𝑈(𝑡)𝑉(𝑡)𝑉(𝑡)𝑈(𝑡)0,𝑡𝑡0. A conjoined solution (𝑈(𝑡),𝑉(𝑡)) of system (1.1) is said to be a conjoined basis of system (1.1) if the rank of the 2𝑛×𝑛 matrix (𝑈(𝑡),𝑉(𝑡))𝑇 is 𝑛.

In 2000, Kumari and Umamaheswaram [1], Yang and Cheng [2], and Wang [3] used the substitution 𝑊1(𝑥)=𝑎(𝑥)𝑉(𝑥)𝑈1(𝑥)+𝑓(𝑥)𝐸𝑛,𝑎(𝑥)=exp2𝑥𝑥0𝑓(𝑠)𝑑𝑠,(1.2) to study the oscillation of system (1.1). One of the main results in [1] is as follows.

Theorem A. Let 𝐷={(𝑥,𝑠)𝑥0𝑠𝑥} and 𝐷0={(𝑥,𝑠)𝑥0𝑠<𝑥}. Let the functions 𝐻𝐶(𝐷,) and 𝐶(𝐷0,) satisfy the following three conditions: (i)𝐻(𝑥,𝑥)=0, for 𝑥𝑥0,𝐻(𝑥,𝑠)>0 on 𝐷0; (ii)𝐻 has a continuous and nonpositive partial derivative on 𝐷0 with respect to the second variable; (iii)(𝜕/𝜕𝑠)𝐻(𝑥,𝑠)=(𝑥,𝑠)𝐻(𝑥,𝑠), for all (𝑥,𝑠)𝐷0.
If there exists a function 𝑓𝐶1[𝑥0,) such that limsup𝑥1𝐻𝑥,𝑥0𝜆1𝑥𝑥0{𝐻(𝑥,𝑠)𝑇(𝑠)+𝐹(𝑥,𝑠)}𝑑𝑠=,(1.3) where 𝑇(𝑥)=𝑎(𝑥)[𝐶𝑓(𝐴+𝐴)+𝑓2𝐵𝑓𝐸𝑛](𝑥),𝑎(𝑥)=exp{2𝑥𝑥0𝑓(𝑠)𝑑𝑠},𝐸𝑛 is the 𝑛×𝑛 identity matrix, and 𝐹(𝑥,𝑠)=𝐻(𝑥,𝑠)𝑎𝑓𝐴+𝐴𝑎𝐴𝐵1𝐴1(𝑠)𝑎(𝑠)2(𝑥,𝑠)𝐴𝐻(𝑥,𝑠)+𝑓(𝑠)𝐻(𝑥,𝑠)𝐵1+𝐵1𝐴1(𝑠)𝑎(𝑠)2(𝑥,𝑠)+𝑓(𝑠)𝐵𝐻(x,𝑠)1/2(𝑠)𝑓(𝑠)𝐻(𝑥,𝑠)𝐵(𝑠)2,(1.4) 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 𝑊2(𝑥)=𝑎(𝑥)𝑉(𝑥)𝑈1(𝑥)+𝑓(𝑥)𝐵1(𝑥),𝑎(𝑥)=exp2𝑥𝑥0.𝑓(𝑠)𝑑𝑠(1.5) One of the main results in [4] is as follows.

Theorem B. Let the functions 𝐻𝐶(𝐷,) and 𝐶(𝐷0,) satisfy (i)–(iii) in Theorem A and, for all sufficiently large 𝑠,liminf𝑥𝐻(𝑥,𝑠)1. Assume that there exist a function 𝑓𝐶1[𝑥0,) and a monotone subhomogeneous functional 𝑞 of degree 𝑐 on 𝒮 such that limsup𝑥1𝐻𝑥,𝑥0𝑐𝑞𝑥𝑥0𝐻(𝑥,𝑠)𝑅11(𝑠)+4𝑎(𝑠)2(𝑥,𝑠)𝐵11(𝑠)𝑑𝑠=,(1.6) where 𝑅1(𝑥)=𝜙(𝑥)𝑅(𝑥)𝜙(𝑥),𝐵1(𝑥)=𝜙1(𝑥)𝐵(𝑥)[𝜙(𝑥)]1, 𝜙(𝑥) is a fundamental matrix of the linear equation 𝑣=𝐴(𝑥)𝑣, and 𝐴𝑅(𝑥)=𝑎(𝑥)𝐶𝑓𝐵1+𝐵1𝐴+𝑓2𝐵1𝑓𝐵1(𝑥).(1.7) 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 𝐻, be as in Theorem A, and suppose that 0<inf𝑠𝑡0liminf𝑡𝐻(𝑡,𝑠)𝐻𝑡,𝑡0+.(1.8) If there exist a function 𝑓𝐶1[𝑡0,) and a positive linear functional 𝑔 on such that liminf𝑡1𝐻𝑡,𝑡0𝑡𝑡0𝑔𝐶𝐻(𝑡,𝑠)1+𝐴𝐵11𝐵𝐴+11𝐴(𝑠)𝑑𝑠>,limsup𝑡1𝐻𝑡,𝑡0𝑡𝑡02𝐵(𝑡,𝑠)𝑔11(𝑠)𝑑𝑠<,(1.9) and suppose also that there exists a function 𝑚𝐶[𝑡0,) such that limsup𝑡1𝐻(𝑡,𝑇)𝑡𝑇𝑔𝐶𝐻(𝑡,𝑠)1+𝐴𝐵11𝐵𝐴+11𝐴(1𝑠)42(𝑡,𝑠)𝐵11(𝑠)𝑑𝑠,𝑚(𝑇),(1.10) for all 𝑇𝑡0 and 𝑡0𝑚2+(𝑡)𝑔𝐵11(𝑡)𝑑𝑡=+,(1.11) where 𝑚+(𝑡)=max{𝑚(𝑡),0} and 𝐵1(𝑡),𝐶1(𝑡) 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 𝑊3(𝑡)=𝑎(𝑡)𝑌(𝑡)𝑋1(𝑡)+𝑓(𝑡)𝐵1(𝑡),𝑡𝑡0,(1.12) where 𝑎(𝑡) is as in (1.5). One of the main results in [7] is as follows.

Theorem D. Let 𝐻, be as in Theorem A, and suppose that there exist a function 𝑓𝐶1[𝑡0,) and a positive linear functional 𝑔 on , for some 𝛽1, such that limsup𝑡1𝐻𝑡,𝑡0𝑡𝑡0𝑔𝐶𝐻(𝑡,𝑠)1+𝐴𝐵11𝐵𝐴+11𝐴(𝛽𝑠)42(𝑡,𝑠)𝐵11(𝑠)𝑑𝑠=,(1.13) where 𝐵1(𝑡)=𝑎1(𝑡)𝐵(𝑡),𝑎(𝑡)=exp2𝑡,𝐶𝑓(𝑠)𝑑𝑠1(𝐵𝑡)=𝑎(𝑡)𝐶(𝑡)+𝑓(𝑡)1𝐴+𝐴𝐵1(𝑡)+𝑓(𝑡)𝐵1(𝑡)𝑓2(𝑡)𝐵1(.𝑡)(1.14) 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 𝐷={(𝑡,𝑠)𝑡0<𝑠𝑡<+} and 𝐷0={(𝑡,𝑠)𝑡0<𝑠<𝑡<+}. We say that a continuous function 𝐻(𝑡,𝑠)𝐷+ belongs to the class if (i)𝐻(𝑡,𝑡)=0 for 𝑡𝑡0, 𝐻(𝑡,𝑠)>0 on 𝐷0, (ii)𝐻 has a continuous and nonpositive partial derivative on 𝐷0 with respect to the second variable, (iii)(𝜕/𝜕𝑠)(𝐻(𝑡,𝑠)𝑘(𝑠))=(𝑡,𝑠)𝐻(𝑡,𝑠)𝑘(𝑠), for all (𝑡,𝑠)𝐷0, where 𝑘(𝑡)𝐶1([𝑡0,+),(0,+)).

We now follow [8] in defining the space 𝒮 as the real linear spare of all real symmetric 𝑛×𝑛 matrices. Let 𝑔 be a linear functional on , 𝑔 is said to be positive if 𝑔(𝐴)>0 whenever 𝐴𝒮 and 𝐴>0.

2. Main Results

In this paper, we need the following lemma.

Lemma 2.1 (see [6]). If 𝑔 is a positive linear functional on , then, for all 𝐴,𝐵, one has ||𝑔𝐴𝐵||2𝐴𝑔𝐴𝑔𝐵𝐵.(2.1)

Theorem 2.2. Let 𝐻(𝑡,𝑠). If there exist a function 𝑏𝐶1([𝑡0,+),+), a matrix function 𝜓𝐶1([𝑡0,+),𝒮), and a positive linear functional 𝑔 on , for some 𝛼1, such that limsup𝑡+1𝐻𝑡,𝑡0𝑡𝑡0𝑔𝐻(𝑡,𝑠)𝑘(𝑠)𝑇1(𝛼𝑠)4𝐵11(𝑠)𝑇2(𝑡,𝑠)2𝑑𝑠=,(2.2) where 𝐵1(𝑡)=(1/𝑏(𝑡))𝐵(t),𝐷(𝑡)=𝐴(𝑡)𝑏(𝑡)𝐵1(𝑡)𝜓(𝑡),𝐹1(𝑠)=𝑏(𝑠)[𝐶+𝐴𝜓+𝜓𝐴𝜓𝐵𝜓+𝜓](𝑠),𝑇1(𝑠)=[𝐹1+(𝐵11𝐷)+𝐷𝐵11𝐷(𝑏/𝑏)𝐵11𝐷](𝑠), and 𝑇2(𝑡,𝑠)=(𝑡,𝑠)𝐻(𝑡,𝑠)𝑘(𝑠)(𝑏(𝑠)/𝑏(𝑠)), 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 (𝑈(𝑡),𝑉(𝑡)) such that 𝑈(𝑡) is nonsingular for all sufficiently large 𝑡. Without loss of generality, we assume that det𝑈(𝑡)0 for all 𝑡𝑡0. This allows us to make a Riccati transformation 𝑊(𝑡)=𝑏(𝑡)𝑉(𝑡)𝑈1,(𝑡)+𝜓(𝑡)(2.3) for all 𝑡𝑡0. Then, 𝑊(𝑡) is well defined, Hermitian, and solves the Riccati equation 𝑊𝑏(𝑡)(𝑡)𝑊𝑏(𝑡)(𝑡)+𝑊(𝑡)(𝐴𝐵𝜓)(𝑡)+(𝐴𝐵𝜓)1(𝑡)𝑊(𝑡)𝑊𝑏(𝑡)(𝑡)𝐵(𝑡)𝑊(𝑡)+𝐹1(𝑡)=0,(2.4) on [𝑡0,).
Let 𝐵1(𝑡)=(1/𝑏(𝑡))𝐵(𝑡),𝐷(𝑡)=𝐴(𝑡)𝑏(𝑡)𝐵1(𝑡)𝜓(𝑡). So, from (2.4), we have 𝑊𝑏(𝑡)(𝑡)𝑊𝑏(𝑡)(𝑡)+𝑊(𝑡)𝐷(𝑡)+𝐷(𝑡)𝑊(𝑡)𝑊(𝑡)𝐵1(𝑡)𝑊(𝑡)+𝐹1(𝑡)=0.(2.5) Now by the substitution 𝑃(𝑡)=𝑊(𝑡)𝐵11(𝑡)𝐷(𝑡) in (2.5), we obtain 𝑃𝑏(𝑡)(𝑡)𝑃𝑏(𝑡)(𝑡)𝑃(𝑡)𝐵1(𝑡)𝑃(𝑡)+𝑇1(𝑡)=0.(2.6) By rearranging the terms, we get 𝑇1(𝑡)=𝑃𝑏(𝑡)+(𝑡)𝑃𝑏(𝑡)(𝑡)+𝑃(𝑡)𝐵1(𝑡)𝑃(𝑡).(2.7) Multiplying (2.7), with 𝑡 replaced by 𝑠, by 𝐻(𝑡,𝑠)𝑘(𝑠) and integrating from 𝑡0 and 𝑡, we obtain 𝑡𝑡0𝐻(𝑡,𝑠)𝑘(𝑠)𝑇1(=𝑠)𝑑𝑠𝑡𝑡0𝑃𝐻(𝑡,𝑠)𝑘(𝑠)(𝑏𝑠)(𝑠)𝑏(𝑠)𝑃(𝑠)𝑃(𝑠)𝐵1(𝑠)𝑃(𝑠)𝑑𝑠=𝐻𝑡,𝑡0𝑘𝑡0𝑃𝑡0+𝑡𝑡0𝑃(𝑠)(𝑡,𝑠)𝑏𝐻(𝑡,𝑠)𝑘(𝑠)𝐻(𝑡,𝑠)𝑘(𝑠)(𝑠)𝑏(𝑠)𝑑𝑠𝑡𝑡0𝐻(𝑡,𝑠)𝑘(𝑠)𝑃(𝑠)𝐵1(𝑠)𝑃(𝑠)𝑑𝑠.(2.8) Taking the linear functional 𝑔 on both sides of the above equation, we have, for some 𝛼1, 𝑡𝑡0𝑔𝐻(𝑡,𝑠)𝑘(𝑠)𝑇1(𝑠)𝑑𝑠=𝐻𝑡,𝑡0𝑘𝑡0𝑔𝑃𝑡0+𝑡𝑡0𝑔[]𝑃(𝑠)(𝑡,𝑠)𝑏𝐻(𝑡,𝑠)𝑘(𝑠)𝐻(𝑡,𝑠)𝑘(𝑠)(𝑠)𝑏(𝑠)𝑑𝑠𝑡𝑡0𝑃𝐻(𝑡,𝑠)𝑘(𝑠)𝑔(𝑠)𝐵1(𝑠)𝑃(𝑠)𝑑𝑠𝐻𝑡,𝑡0𝑘𝑡0𝑔𝑃𝑡0+𝑡𝑡0𝑔[]𝑃(𝑠)(𝑡,𝑠)𝑏𝐻(𝑡,𝑠)𝑘(𝑠)𝐻(𝑡,𝑠)𝑘(𝑠)(𝑠)𝑏(𝑠)𝑑𝑠𝑡𝑡0𝑔𝐵𝐻(𝑡,𝑠)𝑘(𝑠)11(𝑠)1[]}{𝑔𝑃(𝑠)2𝑑𝑠=𝐻𝑡,𝑡0𝑘𝑡0𝑔𝑃𝑡0𝑡𝑡0𝐻(𝑡,𝑠)𝑘(𝑠)𝛼𝑔[𝐵11𝑔[](𝑠)]𝑃(𝑠)𝐵𝛼𝑔11(𝑠)2(𝑡,s)𝑏𝐻(𝑡,𝑠)𝑘(𝑠)(𝑠)𝑏(𝑠)2+𝛼𝑑𝑠4𝑡𝑡0𝑔𝐵11(𝑠)(𝑡,𝑠)𝐻𝑏(𝑡,𝑠)𝑘(𝑠)(𝑠)𝑏(𝑠)2𝑑𝑠𝛼1𝛼𝑡𝑡0𝐻𝑔𝐵(𝑡,𝑠)𝑘(𝑠)11(𝑠)1[𝑃]}{𝑔(𝑠)2𝑑𝑠𝐻𝑡,𝑡0𝑘𝑡0𝑔𝑃𝑡0+𝛼4𝑡𝑡0𝑔𝐵11𝑇(𝑠)2(𝑡,𝑠)2𝑑𝑠.(2.9) So, 𝑡𝑡0𝑔𝐻(𝑡,𝑠)𝑘(𝑠)𝑇1(𝛼𝑠)4𝐵11(𝑠)𝑇2(𝑡,𝑠)2𝑑𝑠𝐻𝑡,𝑡0𝑘𝑡0𝑔𝑃𝑡0.(2.10) Taking the upper limit in both sides of (2.10) as 𝑡, we obtain limsup𝑡1𝐻𝑡,𝑡0𝑡𝑡0𝑔𝐻(𝑡,𝑠)𝑘(𝑠)𝑇1(𝛼𝑠)4𝐵11(𝑠)𝑇2(𝑡,𝑠)2𝑡𝑑𝑠𝑘0𝑔𝑃𝑡0,(2.11) which contradicts (2.2). This completes the proof of Theorem 2.2.

Theorem 2.3. Let the functions 𝐻, and 𝑏,𝑔 be as in Theorem 2.2, and suppose that 0<infs𝑡0liminf𝑡𝐻(𝑡,𝑠)𝐻𝑡,𝑡0+.(2.12) If there exists a function 𝜙𝐶([𝑡0,)), such that, for all 𝑡𝑇𝑡0, and for some 𝛼1, limsup𝑡1𝐻(𝑡,𝑇)𝑡𝑇𝑔𝐻(𝑡,𝑠)𝑘(𝑠)𝑇1(𝛼𝑠)4𝐵11(𝑠)𝑇2(𝑡,𝑠)2𝑑𝑠𝜙(𝑇),(2.13)𝑡0𝜙2+(𝑡)𝑔𝐵11(𝑘𝑡)2(𝑡)𝑑𝑡=+,(2.14) where 𝜙+(𝑡)=max{𝜙(𝑡),0},𝐵1(𝑡),𝐹1(𝑡),𝐷(𝑡),𝑇1(𝑡), and 𝑇2(𝑡,𝑠) 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 𝑡𝑇𝑡0, and for some 𝛼1, 1𝐻(𝑡,𝑇)𝑡𝑇𝑔𝐻(𝑡,𝑠)𝑘(𝑠)𝑇1(𝛼𝑠)4𝐵11(𝑠)𝑇2(𝑡,𝑠)2[]𝑑𝑠𝑘(𝑇)𝑔𝑃(𝑇)𝛼1𝛼1𝐻(𝑡,𝑇)𝑡𝑇𝑔𝐵𝐻(𝑡,𝑠)𝑘(𝑠)11(𝑠)1[]}{𝑔𝑃(𝑠)2𝑑𝑠.(2.15) Taking the upper limit of the above inequation as 𝑡, limsup𝑡1𝐻(𝑡,𝑇)𝑡𝑇𝑔𝐻(𝑡,𝑠)𝑘(𝑠)𝑇1(𝛼𝑠)4𝐵11(𝑠)𝑇2(𝑡,𝑠)2[]𝑑𝑠𝑘(𝑇)𝑔𝑃(𝑇)𝛼1𝛼liminf𝑡1𝐻(𝑡,𝑇)𝑡𝑇𝑔𝐵𝐻(𝑡,𝑠)𝑘(𝑠)11(𝑠)1[]}{𝑔𝑃(𝑠)2𝑑𝑠.(2.16) By (2.13), we obtain []𝑘(𝑇)𝑔𝑃(𝑇)𝜙(𝑇)+𝛼1𝛼liminf𝑡1𝐻(𝑡,𝑇)𝑡𝑇𝑔𝐵𝐻(𝑡,𝑠)𝑘(𝑠)11(𝑠)1{𝑔[]}𝑃(𝑠)2𝑑𝑠,(2.17)[]𝑘(𝑇)𝑔𝑃(𝑇)𝜙(𝑇).(2.18) Besides, we have liminf𝑡1𝐻𝑡,𝑡0𝑡𝑡0𝑔𝐵𝐻(𝑡,𝑠)𝑘(𝑠)11(𝑠)1{𝑔[]}𝑃(𝑠)2𝛼𝑑𝑠𝜙𝑡𝛼10𝑡+𝑘0𝑔𝑃𝑡0<.(2.19) Now, we claim that 𝑡+0𝑔𝐵11(𝑠)1[𝑃]}{𝑔(𝑠)2𝑑𝑠<.(2.20) Suppose to the contrary that 𝑡+0𝑔𝐵11(𝑠)1[𝑃]}{𝑔(𝑠)2𝑑𝑠=+.(2.21) By (2.12), there exists a positive constant 𝜀 satisfying inf𝑠𝑡0liminf𝑡𝐻(𝑡,𝑠)𝐻𝑡,𝑡0>𝜀>0.(2.22) And according to the above 𝜀, there exists 𝑡1𝑡0 such that 𝑡𝑡0𝑔𝐵11(𝑠)1{𝑔[]}𝑃(𝑠)21𝑑𝑠>𝜀2,𝑡𝑡1.(2.23) Thus, 1𝐻𝑡,𝑡0𝑡𝑡0𝑔𝐵𝐻(𝑡,𝑠)𝑘(𝑠)11(𝑠)1{𝑔[]}𝑃(𝑠)2=1𝑑𝑠𝐻𝑡,𝑡0𝑡𝑡0𝐻(𝑡,𝑠)𝑘(𝑠)𝑑𝑠𝑡0𝑔𝐵11(𝜉)1[]}{𝑔𝑃(𝜉)21𝑑𝜉=𝐻𝑡,𝑡0𝑡𝑡0𝜕(𝐻(𝑡,𝑠)𝑘(𝑠))𝜕𝑠𝑠𝑡0𝑔𝐵11(𝜉)1[]}{𝑔𝑃(𝜉)2>1𝑑𝜉𝑑𝑠𝜀21𝐻𝑡,𝑡0𝑡𝑡1𝜕(𝐻(𝑡,𝑠)𝑘(𝑠))=𝑘𝑡𝜕𝑠𝑑𝑠1𝜀2𝐻𝑡,𝑡1𝐻𝑡,𝑡0.(2.24) From (2.22), there exists a 𝑡2𝑡1 such that, for all 𝑡𝑡2, 𝐻𝑡,𝑡1𝐻𝑡,𝑡0>𝜀.(2.25) So, 1𝐻𝑡,𝑡0𝑡𝑡0𝑔𝐵𝐻(𝑡,𝑠)𝑘(𝑠)11(𝑠)1[]}{𝑔𝑃(𝑠)2𝑘𝑡𝑑𝑠>1𝜀.(2.26) Since 𝜀 is arbitrary, we get liminf𝑡1𝐻𝑡,𝑡0𝑡𝑡0𝑔𝐵𝐻(𝑡,𝑠)𝑘(𝑠)11(𝑠)1{𝑔[]}𝑃(𝑠)2𝑑𝑠=,(2.27) which contradicts (2.19). So, (2.20) holds; then, by (2.18) and (2.20), we can obtain 𝑡0𝜙2+(𝑡)𝑔𝐵11𝑘(𝑡)2(𝑡)𝑑𝑡𝑡0𝑔[]𝑃(𝑡)2𝑔𝐵11(𝑡)𝑑𝑡<,(2.28) which contradicts (2.14). This completes our proof of Theorem 2.3.

Example 2.4. Consider the linear Hamiltonian system (1.1), where 𝐵(𝑡)=𝑡𝐼2,𝐶(𝑡)=((1/𝑡)cos𝑡+(3/4𝑡3))𝐼2,𝐴(𝑡)=01/𝑡1/𝑡0 are 2×2-matrices and 𝐵,𝐶 are Hermitian.
Let 𝐻(𝑡,𝑠)=(𝑡𝑠)2,(𝑡,𝑠)=2,𝑏(𝑡)=𝑡,𝜓(𝑡)=(1/2𝑡2)𝐼2, and 𝑔[𝐴]=𝑎11, where 𝐴=(𝑎𝑖𝑗) is a 2×2-matrix. Then, lim𝑡(𝐻(𝑡,𝑠)/𝐻(𝑡,𝑡0))=(𝑡𝑠)2/(𝑡𝑡0)2=1,𝐵1(𝑡)=𝐼2, 𝐷(𝑡)=1/2𝑡1/𝑡1/𝑡1/2𝑡, 𝐹1(𝑡)=cos𝑡𝐼2,𝑇(𝑡)=1/4𝑡2cos𝑡2/𝑡22/𝑡21/4𝑡2cos𝑡, limsup𝑡(1/𝑡2)𝑡𝑇g{(𝑡𝑠)2𝑇(𝑠)(𝛼/4)𝐵11(𝑠)[2(𝑡𝑠)(1/𝑠)]2}𝑑𝑠>1/𝑇𝜙(𝑇), and 𝑡0(𝜙2+(𝑡)/𝑔[𝐵11(𝑡)]𝑘2(𝑡))𝑑𝑡=𝑡0(1/𝑡)𝑑𝑡=. According to Theorem 2.3, we get that this linear system is oscillatory.

Remark 2.5. In Theorem 2.2, let 𝑏(𝑡)=exp{2𝑡𝑓(𝑠)𝑑𝑠},𝜓(𝑡)=𝑓(𝑡)𝐵1(𝑡), 𝑘(𝑡)=1. 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 𝐻(𝑡,𝑠),𝑘(𝑠), and 𝑓(𝑠). For example, we can obtain some useful oscillation criteria if we choose 𝐻(𝑡,𝑠)=(𝑥𝑠)𝑚,[ln(𝑥/𝑠)]𝑚,[𝑥𝑠𝑑𝑧/𝜃(𝑧)]𝑚, or 𝜌(𝑥𝑠), 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 [𝑡0,). We assume that a function 𝐻=𝐻(𝑡,𝑠) satisfying (i). Further, we assume that 𝑘(𝑡)=1 and 𝐻(𝑡,𝑠) has partial derivatives 𝜕𝐻/𝜕𝑡 and 𝜕𝐻/𝜕𝑠 on 𝐷 such that 𝜕𝜕𝑡𝐻(𝑡,𝑠)=1(𝑡,𝑠)𝜕𝐻(𝑡,𝑠),(3.1)𝜕𝑠𝐻(𝑡,𝑠)=2(𝑡,𝑠)𝐻(𝑡,𝑠),(3.2) where 1,2𝐿loc(𝐷,).

We first prove two lemmas.

Lemma 3.1. Suppose that (𝑈(𝑡),𝑉(𝑡)) is a nontrivial prepared solution of system (1.1) such that det𝑈(𝑡)0 on (𝑎1,𝑎2][𝑡0,). Then, for any 𝑏(𝑡)𝐶1([𝑡0,),+), matrix function 𝜓𝐶1([𝑡0,),𝒮), 𝐻 satisfies (i), (3.1) and (3.2), and a positive linear functional 𝑔 on , one has, for some 𝛼1, 1𝐻𝑎2,𝑎1𝑎2𝑎1𝑔𝐻𝑡,𝑎1𝑇1𝛼(𝑠)4𝐵11(𝑡)1𝑡,𝑎1+𝐻𝑡,𝑎1𝑏(𝑡)𝑏(𝑡)2𝑃𝑎𝑑𝑡𝑔2,(3.3) where 𝑊(𝑡) is defined by (2.3) on (𝑎1,𝑎2], 𝐵1(𝑡),𝐷(𝑡),𝐹1(𝑠), and 𝑇1(𝑠) are the same as in Theorem 2.2.

Proof. Since (𝑈(𝑡),𝑉(𝑡)) is a nontrivial prepared solution of system (1.1) such that 𝑈(𝑡) is nonsingular on (𝑎1,𝑎2], then, 𝑊(𝑡) by (2.3) is well defined and solves the Riccati equation (2.7) on (𝑎1,𝑎2].
On multiplying (2.7) by 𝐻(𝑡,𝑠) and integrating with respect to 𝑡 from 𝑠 to 𝑎2 for 𝑠(𝑎1,𝑎2], we can find 𝑎2𝑠𝐻(𝑡,𝑠)𝑇1=(𝑡)𝑑𝑡𝑎2𝑠𝐻(𝑡,𝑠)𝑃(𝑡)𝑑𝑡𝑎2𝑠𝑏𝐻(𝑡,𝑠)(𝑡)𝑏(𝑡)𝑃(𝑡)𝑑𝑡𝑎2𝑠𝐻(𝑡,𝑠)𝑃(𝑡)𝐵1𝑎(𝑡)𝑃(𝑡)𝑑𝑡=𝐻2𝑃𝑎,𝑠2𝑎2𝑠𝑃(𝑡)1(𝑡,𝑠)𝑏𝐻(𝑡,𝑠)+𝐻(𝑡,𝑠)(𝑡)𝑏(𝑡)𝑑𝑡𝑎2𝑠𝐻(𝑡,𝑠)𝑃(𝑡)𝐵1(𝑡)𝑃(𝑡)𝑑𝑡.(3.4) Taking the linear functional 𝑔 on both sides of the above equation, we have, for some 𝛼1, 𝑎2𝑠𝑔𝐻(𝑡,𝑠)𝑇1𝑎(𝑡)𝑑𝑡=𝐻2𝑔𝑃𝑎,𝑠2𝑎2𝑠𝑔[]𝑃(𝑡)1(𝑡,𝑠)𝑏𝐻(𝑡,𝑠)+𝐻(𝑡,𝑠)(𝑡)𝑏(𝑡)𝑑𝑡𝑎2𝑠𝑃𝐻(𝑡,𝑠)𝑔(𝑡)𝐵1𝑎(𝑡)𝑃(𝑡)𝑑𝑡𝐻2𝑔𝑃𝑎,𝑠2𝑎2𝑠𝑔[]𝑃(𝑡)1(𝑡,𝑠)𝑏𝐻(𝑡,𝑠)+𝐻(𝑡,𝑠)(𝑡)𝑏(𝑡)𝑑𝑡𝑎2𝑠𝑔𝐵𝐻(𝑡,𝑠)11(𝑡)1{𝑔[]}𝑃(𝑡)2𝑎𝑑𝑡=𝐻2𝑔𝑃𝑎,𝑠2𝑎2𝑠𝐻(𝑡,𝑠)𝛼𝑔[𝐵11(𝑡)]𝑔[𝑃(𝑡)]+𝐵𝛼𝑔11(𝑡)21(𝑡,𝑠)+𝑏𝐻(𝑡,𝑠)(𝑡)𝑏(𝑡)2+𝛼𝑑𝑡4𝑎2𝑠𝑔𝐵11(𝑡)1(𝑡,𝑠)+𝑏𝐻(𝑡,𝑠)(𝑡)𝑏(𝑡)2𝑑𝑡𝛼1𝛼𝑎2𝑠𝐻𝑔𝐵(𝑡,𝑠)11(𝑡)1[𝑃]}{𝑔(𝑡)2𝑎𝑑𝑡𝐻2𝑔𝑃𝑎,𝑠2+𝛼4𝑎2𝑠𝑔𝐵11(𝑡)1(𝑡,𝑠)+𝑏𝐻(𝑡,𝑠)(𝑡)𝑏(𝑡)2𝑑𝑡.(3.5) That is, 1𝐻𝑎2,𝑠𝑎2𝑠𝑔𝐻(𝑡,𝑠)𝑇1𝛼(𝑡)4𝐵11(𝑡)1(𝑡,𝑠)+𝑏𝐻(𝑡,𝑠)(𝑡)𝑏(𝑡)2𝑃𝑎𝑑𝑡𝑔2.(3.6) Let 𝑠𝑎1, 1𝐻𝑎2,𝑎1𝑎2𝑎1𝑔𝐻𝑡,𝑎1𝑇1𝛼(𝑡)4𝐵11(𝑡)1𝑡,𝑎1+𝐻𝑡,𝑎1𝑏(𝑡)𝑏(𝑡)2𝑃𝑎𝑑𝑡𝑔2.(3.7)

Lemma 3.2. Suppose that (𝑈(𝑡),𝑉(𝑡)) is a nontrivial prepared solution of system (1.1) such that det𝑈(𝑡)0 on (𝑎2,𝑎3][𝑡0,). Then, for any 𝑏(𝑡)𝐶1([𝑡0,),+), matrix function 𝜓𝐶1([𝑡0,),𝒮), 𝐻 satisfies (i), (3.1) and (3.2), and a positive linear functional 𝑔 on , one has, for some 𝛼1, 1𝐻𝑎3,𝑎2𝑎3𝑎2𝑔𝑎𝐻3𝑇,𝑠1𝛼(𝑠)4𝐵11(𝑠)1𝑎3+,𝑠𝐻𝑎3𝑏,𝑠(𝑠)𝑏(𝑠)2𝑃𝑎𝑑𝑠𝑔2,(3.8) where 𝑊(𝑡) is defined by (2.3) on (𝑎2,𝑎3], 𝐵1(𝑡),𝐷(𝑡),𝐹1(𝑠), and 𝑇1(𝑠) are the same as in Theorem 2.2.

Proof. Since (𝑈(𝑡),𝑉(𝑡)) is a nontrivial prepared solution of system (1.1) such that 𝑈(𝑡) is nonsingular on (𝑎2,𝑎3], then, 𝑊(𝑡) by (2.3) is well defined and solves the Riccati equation (2.7) on (𝑎2,𝑎3].
On multiplying (2.7) by 𝐻(𝑡,𝑠), integrating with respect to 𝑠 from 𝑎2 to 𝑡 for 𝑡(𝑎2,𝑎3], and following the proof of Lemma 3.1, we can find 1𝐻𝑡,𝑎2𝑡𝑎2𝑔𝐻(𝑡,𝑠)𝑇1𝛼(𝑠)4𝐵11(𝑠)1(𝑡,𝑠)+𝑏𝐻(𝑡,𝑠)(𝑠)𝑏(𝑠)2𝑃𝑎𝑑𝑠𝑔2<+.(3.9) Let 𝑡𝑎3, 1𝐻𝑎3,𝑎2𝑎3𝑎2𝑔𝑎𝐻3𝑇,𝑠1𝛼(𝑠)4𝐵11(𝑠)1𝑎3+,𝑠𝐻𝑎3𝑏,𝑠(𝑠)𝑏(𝑠)2𝑃𝑎𝑑𝑠𝑔2.(3.10)

Theorem 3.3. Suppose that there exist some 𝑎2(𝑎1,𝑎3)[𝑡0,),𝑏(𝑡)𝐶1([𝑡0,),+), matrix function 𝜓𝐶1([𝑡0,),𝒮), 𝐻 satisfies (i), (3.1) and (3.2), and a positive linear functional 𝑔 on such that, for some 𝛼1, 𝑔1𝐻𝑎2,𝑎1𝑎2𝑎1𝐻𝑡,𝑎1𝑇1𝛼(𝑡)4𝐵11(𝑡)1𝑡,𝑎1+𝐻𝑡,𝑎1𝑏(𝑡)𝑏(𝑡)2+1𝑑𝑡𝐻𝑎3,𝑎2𝑎3𝑎2𝑎𝐻3𝑇,𝑠1𝛼(𝑠)4𝐵11(𝑠)1𝑎3+,𝑠𝐻𝑎3𝑏,𝑠(𝑠)𝑏(𝑠)2𝑑𝑠>0,(3.11) where 𝐵1(𝑡),𝐷(𝑡),𝐹1(𝑠), and 𝑇1(𝑠) are defined as in Theorem 2.2. Then, for any nontrivial prepared solution (𝑈(𝑡),𝑉(𝑡)) of system (1.1), det𝑈(𝑡) has at least one zero in (𝑎1,𝑎3).

Theorem 3.4. If, for each 𝑇𝑡0, there exist 𝑏(𝑡)𝐶1([𝑡0,),+), matrix function 𝜓𝐶1([𝑡0,),𝒮), 𝐻 satisfies (i), (3.1), (3.2), a positive linear functional 𝑔 on and 𝑎1,𝑎2,𝑎3, such that 𝑇𝑎1<𝑎2<𝑎3 and condition (3.1) holds, where 𝐵1(𝑡),𝐷(𝑡),𝐹1(𝑠), and 𝑇1(𝑡) are defined as in Theorem 2.2, then, system (1.1) is oscillatory.

In conclusion, we note that the results given here can extend, improve and complement Theorems A–D, and deal with some cases not covered by known criteria by choosing the functions 𝐻,𝑏,𝜙, and 𝑔. From our results, we can derive a number of easily verifiable oscillation criteria.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (11171178), the National Ministry of Education under Grant (20103705110003), and the Natural Sciences Foundation of Shandong Province under Grant (ZR2009AM011).

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Copyright © 2012 Nan Wang 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.


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