New Interval Oscillation Criteria for Certain Linear Hamiltonian Systems

Abstract

Using a generalized Riccati transformation and the general integral means technique, some new interval oscillation criteria for the linear matrix Hamiltonian system 𝑈^′=(𝐴(𝑡)−𝜆(𝑡)𝐼)𝑈+𝐵(𝑡)𝑉, 𝑉^′=𝐶(𝑡)𝑈+(𝜇(𝑡)𝐼−𝐴^∗(𝑡))𝑉, 𝑡≥𝑡₀ 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: 𝑈^′=(𝐴(𝑡)(𝑡)𝐼)(𝑡)𝑉−𝜆𝑈+𝐵𝑉,_′=𝐶(𝑡)𝑈+𝜇(𝑡)𝐼−𝐴^∗(𝑡)𝑉, (1) where 𝐴(𝑡), 𝐵(𝑡), and 𝐶(𝑡) are 𝑛×𝑛 matrices of real valued continuous functions on the interval, 𝐽=[𝑡₀,∞), 𝐵(𝑡)=𝐵^∗(𝑡), 𝐵(𝑡) is positive definite, 𝐶(𝑡)=𝐶^∗(𝑡), and 𝜇(𝑡), 𝜆(𝑡) are real valued continuous functions on 𝐽. Here and in the sequel, the transpose of matrix 𝑀 is denoted by 𝑀^∗ and its positive definiteness is denoted by 𝑀>0. For any 𝑛×𝑛 Hermitian matrix 𝑀, its eigenvalues are real numbers, and we always denote them by 𝜆₁[𝑀]≥𝜆₂[𝑀]≥⋯≥𝜆_𝑛[𝑀]. The trace of 𝑀 is denoted by tr(𝑀), and recall that tr(𝑀)=∑^𝑛_𝑘=1𝜆_𝑘(𝑀).

When 𝜆(𝑡)=𝜇(𝑡)=0, the system (1) reduces to the following linear self-adjoint matrix Hamiltonian system: 𝑈^′=𝐴₁(𝑡)𝑈+𝐵₁(𝑡)𝑉𝑉,_′=𝐶₁(𝑡)𝑈−𝐴^∗₁(𝑡)𝑉,𝑡∈𝐽. (2)

A solution (𝑈(𝑡),𝑉(𝑡)) of the system (1) (or (2)) is said to be nontrivial if det𝑈(𝑡)≠0 for at least one 𝑡∈𝐽. A nontrivial solution (𝑈(𝑡),𝑉(𝑡)) of the system (1) (or (2)) is said to be prepared or self-conjugate if 𝑈^∗(𝑡)𝑉(𝑡)−𝑉^∗(𝑡)𝑈(𝑡)=0, 𝑡∈𝐽. System (1) (or (2)) is said to be oscillatory on 𝐽 if there is a nontrivial prepared solution (𝑈(𝑡),𝑉(𝑡)) of (1) such that det𝑈(𝑡) vanishes at least once on [𝑇,∞) for each 𝑇≥𝑡₀. Otherwise, it is said to be nonoscillatory.

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

Definition 1. Let ℳ be the linear space of 𝑛×𝑛 real matrices, and let 𝒮 be the subspace of all symmetric matrices in ℳ. For any 𝐴,𝐵∈𝒮, 𝐴≥𝐵 means that 𝐴−𝐵≥0 is positive semidefinite and 𝐴>𝐵 means that 𝐴−𝐵>0 is positive definite.

Definition 2. A function 𝐻=𝐻(𝑡,𝑠) is called to belong to a function class ℋ, denoted by 𝐻∈ℋ, if 𝐻∈𝐶(𝐃,𝐑₊), where 𝐃={(𝑡,𝑠)∶𝑡>𝑠≥𝑡₀} (𝐃 is the closure of 𝐃) which satisfies 𝐻(𝑡,𝑡)=0, 𝐻(𝑡,𝑠)>0 for 𝑡>𝑠≥𝑡₀. Furthermore, 𝐻 has continuous derivatives 𝜕𝐻/𝜕𝑡 and 𝜕𝐻/𝜕𝑠 on 𝐃 such that for all (𝑡,𝑠)∈𝐃, 𝜕𝐻(𝑡,𝑠)𝜕𝑡=ℎ₁(𝑡,𝑠)𝐻(),()𝑡,𝑠𝜕𝐻𝑡,𝑠𝜕𝑠=−ℎ₂(𝑡,𝑠)𝐻(𝑡,𝑠), (3) where ℎ₁,ℎ₂∈𝐶(𝐃,𝐑) are functions determined by 𝐻(𝑡,𝑠). Note that (𝑡−𝑠)^𝜇 for 𝜇>1; ln(𝑡/𝑠) and 𝐺(𝑡−𝑠) with 𝐺∈𝐶¹(0,∞), 𝐺^′(𝑡)>0 for 𝑡>0, and 𝐺(0)=0 belong to function class ℋ. For given 𝑎,𝑏∈𝐑, 𝑎<𝑏, a function 𝑧(𝑡) is called to belong to a function class 𝑍[𝑎,𝑏], denoted by 𝑧∈𝑍[𝑎,𝑏], if 𝑧(𝑡)∈𝐶[𝑎,𝑏], 𝑧^′(𝑡)∈𝐶(𝑎,𝑏), 𝑧(𝑡)>0 for 𝑡∈(𝑎,𝑏), and 𝑧(𝑎)=𝑧(𝑏)=0. We point out that if 𝐻,𝐻^∼∈ℋ, and we define a function with 𝑧(𝑡)=𝐻^𝜇(𝑏,𝑡)𝐻^∼𝜈(𝑡,𝑎) for 𝑡∈[𝑎,𝑏], where 𝜇,𝜈>1/2 are constants, then 𝑧∈𝑍[𝑎,𝑏].

Definition 3. A nonlinear (and possibly discontinuous) function 𝑞∶𝒮→𝐑 with 𝑞(𝐴)≤0 whenever 𝐴≤0 is called negativity preserving; the class of all such negativity-preserving functionals on 𝒮 is denoted by 𝒩(𝒮).

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 𝐻=𝐻(𝑡,𝑠) as follows.

Theorem 4. Suppose that there exists 𝑓(𝑡)∈𝐶(𝐽;𝐑) such that 𝑓𝐵₁⁻¹ is differentiable, and 𝛼(𝑡)=exp{−2∫^𝑡𝑓(𝑠)𝑑𝑠}. If there exists 𝑞∈𝒩(𝒮), 𝑧∈𝑍[𝑎,𝑏] such that 𝑞^𝑏_𝑎𝑧²𝐷₁−𝛼𝑧𝐴^∗₁−𝑧^′𝐼𝐵₁⁻¹𝑧𝐴₁−𝑧^′𝐼(𝑡)𝑑𝑡>0, (4) where 𝐷₁(𝑡)={𝛼[−𝐶₁−2𝑓𝐾₁+𝑓²𝐵₁⁻¹−(𝑓𝐵₁⁻¹)^′]}(𝑡) and 𝐾₁(𝑡)=(𝐴^∗₁𝐵₁⁻¹+𝐵₁⁻¹𝐴₁)(𝑡). Then system (2) is oscillatory.

Theorem 5. Suppose that 0<𝐵₁⁻¹(𝑡)≤𝐼. If, for each 𝑟≥𝑡₀ and for some 𝜇,𝜈>1, limsup_𝑡→∞1𝑡^{𝜇+𝜈−1}𝜆₁×^𝑡_𝑟(𝑡−𝑠)^𝜇(𝑠−𝑟)^𝜈×𝐷₀(𝑠)+2𝜈𝑡−𝜇+𝜈𝑠+𝜇𝑟()𝐾𝑡−𝑠)(𝑠−𝑟¹(𝑠)ΓΓ()𝑑𝑠>𝜇𝜈𝜇+𝜈−2𝜇−1𝜈−1,4Γ𝜇+𝜈 (5) where 𝐷₀(𝑠)=(−𝐶₁−𝐴^∗₁𝐵₁⁻¹𝐴₁)(𝑠), 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 𝑓(𝑡)∈𝐶(𝐽;𝐑) such that 𝑓𝐵⁻¹ is differentiable, and 𝛼(𝑡)=exp{−2∫^𝑡𝑓(𝑠)𝑑𝑠}. If there exists 𝑞∈𝒩(𝒮), 𝑧∈𝑍[𝑎,𝑏] such that 𝑞^𝑏_𝑎𝑧²𝐷(𝑡)−𝛼(𝑡)𝑒^{−∫𝑡𝑡0(𝜆+𝜇)(𝑠)𝑑𝑠}𝑧𝐴^∗−𝑄𝐵−𝑧^′𝐼𝐵⁻¹×𝑧(𝐴−𝐵𝑄)−𝑧^′𝐼(𝑡)𝑑𝑡>0, (6) where 𝐷(𝑡)=𝛼−𝑒^{−∫(𝜆+𝜇)}𝑄𝐴−𝑄𝐵𝑄+𝐶−𝜆+𝜇𝑄+𝐴^∗𝑄−2𝑓𝐾+𝑒^{−∫(𝜆+𝜇)}𝑓²𝐵⁻¹−𝑒^{−∫(𝜆+𝜇)}𝑓𝐵^−1′𝑡),𝐾(𝑡)(=𝑒^{−∫𝑡𝑡0(𝜆+𝜇)(𝑠)𝑑𝑠}𝐴^∗𝐵⁻¹+𝐵⁻¹𝐴(𝑡)−2𝑄, (7) and 𝑄 is any constant nonzero symmetric 𝑛×𝑛 matrix, then system (1) is oscillatory.

Proof. Suppose that there exists a prepared solution (𝑈(𝑡),𝑉(𝑡)) of (1) which is not oscillatory. Without loss of generality, we may assume that det𝑈(𝑡)≠0 for 𝑡∈𝐽. Define 𝑊(𝑡)=𝛼(𝑡)𝑒^{−∫𝑡𝑡0(𝜆+𝜇)(𝑠)𝑑𝑠}×𝑉(𝑡)𝑈⁻¹(𝑡)+𝑓(𝑡)𝐵⁻¹(𝑡)+𝑄,𝑡∈𝐽. (8) Then 𝑊(𝑡) is Hermitian and satisfies the Riccati equation 𝑊^′(𝑡)1=−𝛼(𝑡)𝑒^{∫𝑡𝑡0(𝜆+𝜇)(𝑠)𝑑𝑠}×𝑊𝐵𝑊−𝐴^∗𝑊−𝑊𝐴+𝑄𝐵𝑊+𝑊𝐵𝑄−𝐷(𝑡). (9) Let 𝐸(𝑡)=𝑊(𝑡)+𝛼(𝑡)𝑒^{−∫𝑡𝑡0(𝜆+𝜇)(𝑠)𝑑𝑠}(𝐵⁻¹𝐴−𝑄)(𝑡). Then we get 1𝛼𝐸^∗1𝐵𝐸=𝛼𝑊^∗𝐵𝑊+𝑒^{−∫(𝜆+𝜇)}𝐴^∗𝑊+𝑊𝐴+𝛼𝑒^{−∫(𝜆+𝜇)}𝐴^∗𝐵⁻¹𝐴−𝑒^{−∫(𝜆+𝜇)}(𝑄𝐵𝑊−𝑊𝐵𝑄)+𝛼𝑒^{−2∫(𝜆+𝜇)}𝑄𝐵𝑄−𝐴^∗𝑄−𝑄𝐴. (10) It follows from (9) and (10) that, for 𝑡∈𝐽, 𝑊^′1=−𝛼𝑒^{−∫(𝜆+𝜇)}𝐸^∗𝐵𝐸−𝐷+𝛼𝑒^{−∫(𝜆+𝜇)}𝐴^∗𝐵⁻¹𝐴−𝛼𝑒^{−∫(𝜆+𝜇)}𝐴𝑄^∗+𝑄𝐴−𝑄𝐵𝑄. (11) Since 𝐵(𝑡)>0, we define 𝑅(𝑡)=[𝑒^{∫(𝜆+𝜇)}𝐵]^1/2(𝑡). Let 𝐺(𝑡)=(𝑧𝑅𝐸𝑅)(𝑡)−𝛼(𝑡)𝑧^′(𝑡)𝐼. Then 1𝛼𝑅⁻¹𝐺^∗𝐺𝑅⁻¹=𝑧²𝛼𝐸^∗𝑒^{∫(𝜆+𝜇)}𝐵𝐸−𝑧𝑧^′𝐸^∗+𝐸+𝛼𝑧^′2𝑒^{∫(𝜆+𝜇)}𝐵⁻¹=𝑧²𝛼𝑒^{∫(𝜆+𝜇)}𝐸^∗𝐵𝐸−𝑧𝑧^′𝐸^∗+𝐸+𝛼𝑒^{−∫(𝜆+𝜇)}𝑧^′2𝐵⁻¹. (12) For each 𝑇≥𝑡₀, by assumptions, there exist 𝑎,𝑏∈𝐑, 𝑇≤𝑎<𝑏, and 𝑧∈𝐶¹[𝑎,𝑏] satisfying 𝑧(𝑎)=𝑧(𝑏)=0. Now, multiplying (11) by 𝑧²(𝑡) and integrating it from 𝑎 to 𝑏, we have ^𝑏_𝑎𝑧²(𝑡)𝐷(𝑡)≤𝑑𝑡_𝑏𝑎𝛼(𝑡)𝑒^{−∫𝑡𝑡0(𝜆+𝜇)(𝑠)𝑑𝑠}×𝑧²𝐴^∗𝐵⁻¹𝐴−𝐴^∗𝑄−𝑄𝐴+𝑄𝐵𝑄−𝑧𝑧^′𝐴^∗𝐵⁻¹+𝐵⁻¹𝐴−2𝑄+𝑧^′2𝐵⁻¹(𝑡)=𝑑𝑡_𝑏𝑎𝛼(𝑡)𝑒^{−∫𝑡𝑡0(𝜆+𝜇)(𝑠)𝑑𝑠}×𝑧(𝐴−𝐵𝑄)^∗−𝑧^′𝐼𝐵⁻¹𝑧(𝐴−𝐵𝑄)−𝑧^′𝐼(𝑡)𝑑𝑡, (13) which implies that ^𝑏_𝑎𝑧²𝐷−𝛼𝑒^{−∫(𝜆+𝜇)}𝑧𝐴^∗−𝑄𝐵−𝑧^′𝐼×𝐵⁻¹𝑧(𝐴−𝐵𝑄)−𝑧^′𝐼(𝑡)𝑑𝑡≤0. (14) Applying the functional 𝑞 to (14), we obtain a contradiction with assumption (6). This completes the proof.

Corollary 7. Suppose that there exists 𝑓(𝑡)∈𝐶(𝐽;𝐑) such that 𝑓𝐵⁻¹ is differentiable, and 𝛼(𝑡)=exp{−2∫^𝑡𝑓(𝑠)𝑑𝑠}. If, for each 𝑇≥𝑡₀, there exist 𝑎,𝑏∈𝐑, 𝑇≤𝑎<𝑏, and 𝑧∈𝑍[𝑎,𝑏] such that 𝜆_𝑖^𝑏_𝑎𝑧²𝐷−𝛼𝑒^{−∫(𝜆+𝜇)}𝑧𝐴^∗−𝑄𝐵−𝑧^′𝐼×𝐵⁻¹𝑧(𝐴−𝐵𝑄)−𝑧^′𝐼(𝑡)𝑑𝑡>0, (15) for some 𝑖∈{1,2,…,𝑛}, where 𝐷(𝑡) is defined as previously mentioned, then system (1) is oscillatory.

Corollary 8. Suppose that there exists 𝑓(𝑡)∈𝐶(𝐽;𝐑) such that 𝑓𝐵⁻¹ is differentiable, and 𝛼(𝑡)=exp{−2∫^𝑡𝑓(𝑠)𝑑𝑠}. If there exists 𝑞∈𝒩(𝒮) and, for each 𝑟≥𝑡₀, there exist 𝐻,𝐻^∼∈ℋ and 𝜇,𝜈>1/2 such that limsup_𝑡→∞𝑞^𝑡_𝑟𝐻^2𝜇(𝑡,𝑠)𝐻^∼2𝜈(𝑠,𝑟)𝐷(𝑠)−𝛼(𝑠)𝑒^{−∫(𝜆+𝜇)}×Ψ^∗(𝑡,𝑠,𝑟)𝐵⁻¹(𝑠)Ψ(𝑡,𝑠,𝑟)𝑑𝑠>0, (16) where 𝐷(𝑠) is defined as that in Theorem 6, and Ψ(𝑡,𝑠,𝑟)=𝐻^𝜇(𝑡,𝑠)𝐻^∼𝜈(𝑠,𝑟)(𝐴−𝐵𝑄)(𝑠)−𝜈𝐻^𝜇(𝑡,𝑠)𝐻^{∼𝜈−1/2}(𝑠,𝑟)ℎ^∼₁(𝑠,𝑟)+𝜇ℎ₂(𝑡,𝑠)𝐻^𝜇−1/2(𝑡,𝑠)𝐻^∼𝜈(𝑠,𝑟), (17) then system (1) is oscillatory.

Let 𝐻(𝑡,𝑠)=𝐻^∼(𝑡,𝑠)=(𝑡−𝑠)², 𝜇=𝜈=1/2; we have the following useful corollary.

Corollary 9. Suppose that there exists 𝑓(𝑡)∈𝐶(𝐽;𝐑) such that, 𝑒^{−∫(𝜆+𝜇)}𝑓𝐵⁻¹ is differentiable, and 𝛼(𝑡)=exp{−2∫^𝑡𝑓(𝑠)𝑑𝑠}. If there exists 𝑞∈𝒩(𝒮) such that for each 𝑟≥𝑡₀, limsup_𝑡→∞𝑞^𝑡_𝑟(𝑡−𝑠)²(𝑠−𝑟)²𝐷(𝑠)−𝛼(𝑠)𝑒^{−∫(𝜆+𝜇)}×Ψ^∗₁(𝑡,𝑠,𝑟)𝐵⁻¹(𝑠)Ψ₁(𝑡,𝑠,𝑟)𝑑𝑠>0, (18) where 𝐷(𝑠) is defined as in Theorem 6, and Ψ₁(𝑡,𝑠,𝑟)=(𝑡−𝑠)(𝑠−𝑟)(𝐴−𝐵𝑄)(𝑠)−(𝑡+𝑟−2𝑠)𝐼, (19) then system (1) is oscillatory.

Theorem 10. Suppose that 0<𝐵⁻¹(𝑡)≤𝑒^{∫(𝜆+𝜇)}𝐼. If, for each 𝑟≥𝑡₀ and for some 𝜇,𝜈>1, limsup_𝑡→∞1𝑡^{𝜇+𝜈−1}𝜆₁^𝑡_𝑟(𝑡−𝑠)^𝜇(𝑠−𝑟)^𝜈×𝐷₀(𝑠)+𝜈𝑡−𝜇+𝜈𝑠+𝜇𝑟2()𝐾(𝑠)ΓΓ()𝑡−𝑠)(𝑠−𝑟𝑑𝑠>𝜇𝜈𝜇+𝜈−2𝜇−1𝜈−1,4Γ𝜇+𝜈 (20) where 𝐷₀(𝑠)=−𝑒^{−∫(𝜆+𝜇)}𝐶+𝐴^∗𝐵⁻¹𝐴−𝜆+𝜇𝑄(𝑠), (21) and 𝐾(𝑠) are defined as in Theorem 6, then system (1) is oscillatory.

Proof. We choose 𝑓(𝑡)≡0, so 𝑎(𝑡)≡1, and 𝐷(𝑡)=𝛼−𝑒^{−∫(𝜆+𝜇)}×𝑄𝐴−𝑄𝐵𝑄+𝐶−𝜆+𝜇𝑄+𝐴^∗𝑄(𝑡). (22) As in the proof of Theorem 6, we obtain (14), with 𝑎 being replaced by 𝑟, 𝑏 replaced by 𝑡, and the integral variable being replaced by 𝑠, that is, ^𝑡_𝑟𝑧²(𝑠)𝐷₀(𝑠)𝑑𝑠≤−^𝑡_𝑟𝑧𝑧^′𝐾(𝑠)𝑑𝑠+^𝑡_𝑟𝑧^′2(𝑠)𝑒^{−∫(𝜆+𝜇)}𝐵⁻¹(𝑠)𝑑𝑠. (23) Suppose that 𝑧(𝑠)=(𝑡−𝑠)^𝜇/2(𝑠−𝑟)^𝜈/2, we get 𝑢^′(𝑠)=𝑧(𝑠)𝜈𝑡−𝜇+𝜈𝑠+𝜇𝑟2().𝑡−𝑠)(𝑠−𝑟 (24) So ^𝑡_𝑟(𝑡−𝑠)^𝜇(𝑠−𝑟)^𝜈𝐷₀(𝑠)+𝜈𝑡−𝜇+𝜈𝑠+𝜇𝑟2()𝐾(𝑠)≤1𝑡−𝑠)(𝑠−𝑟𝑑𝑠4^𝑡_𝑟(𝑡−𝑠)^𝜇−2(𝑠−𝑟)^𝜈−2×𝜈(𝑡−𝑠)−𝜇(𝑠−𝑟)²𝑒^{−∫(𝜆+𝜇)}𝐵⁻¹(𝑠)≤1𝑑𝑠4^𝑡_𝑟𝜈²(𝑡−𝑠)^𝜇(𝑠−𝑟)^𝜈−2𝑑𝑠−2𝜇𝜈^𝑡_𝑟(𝑡−𝑠)^𝜇−1(𝑠−𝑟)^𝜈−1𝑑𝑠+𝜇^2𝑡_𝑟(𝑡−𝑠)^𝜇−2(𝑠−𝑟)^𝜈𝑑𝑠𝐼. (25) As in the proof of [11, Theorem 2.7], we obtain ^𝑡_𝑟(𝑡−𝑠)^𝜇(𝑠−𝑟)^𝜈𝐷₀(𝑠)+𝜈𝑡−𝜇+𝜈𝑠+𝜇𝑟2()𝐾(𝑠)≤1𝑡−𝑠)(𝑠−𝑟𝑑𝑠4(𝑡−𝑟)^{𝜇+𝜈−1}𝜇𝜈²Γ(𝜈−1)Γ𝜇Γ−(𝜈)Γ𝜇𝜇+𝜈2𝜇𝜈ΓΓ+𝜇𝜇+𝜈₂𝜈Γ(𝜈)Γ𝜇−1Γ𝐼=()𝜇𝜇+𝜈𝑡−𝑟^{𝜇+𝜈−1}ΓΓ()𝜇𝜈𝜇+𝜈−2𝜇−1𝜈−14Γ𝜇+𝜈𝐼. (26) Dividing both sides by 𝑡^{𝜇+𝜈−1} and taking the largest eigenvalue, we have limsup_𝑡→∞1𝑡^{𝜇+𝜈−1}𝜆₁×^𝑡_𝑟(𝑡−𝑠)^𝜇(𝑠−𝑟)^𝜈×𝐷₀(𝑠)+𝜈𝑡−𝜇+𝜈𝑠+𝜇𝑟2()𝐾(𝑠)ΓΓ()𝑡−𝑠)(𝑠−𝑟𝑑𝑠≤𝜇𝜈𝜇+𝜈−2𝜇−1𝜈−1.4Γ𝜇+𝜈 (27) 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 𝐴(𝑡)=01𝑡−1−𝑡𝑡,𝐶(𝑡)=𝛾−𝑡0𝑡3cos𝑡+4𝑡³+2𝑡00𝛾+𝑡𝑡3cos𝑡+4𝑡³−2𝑡,𝐵(𝑡)+𝑡=𝑡𝐼₂.,𝑡≥1,𝜆=1,𝜇=−1,𝑄=0110 (28) We get that 𝐾(𝑡)=0. Let 𝑓(𝑡)=−1/2𝑡; let 𝛼(𝑡)=𝑡, and we have 𝐷(𝑡)=𝛾cos𝑡𝐼₂. For each 𝑇≥1, select 𝑘 to be large enough such that 𝑇<(−𝜋/2+2𝑘𝜋)=𝑎<(𝜋/2+2𝑘𝜋)=𝑏. Let 𝑢(𝑡)=sin𝑡. Using Theorem 6, we have ^𝑏_𝑎𝑧²𝐷−𝛼𝑒^{−∫(𝜆+𝜇)}𝑧(𝐴−𝐵𝑄)^∗−𝑧^′𝐼×𝐵⁻¹𝑧(𝐴−𝐵𝑄)−𝑧^′𝐼𝑡)=(𝑑𝑡_{𝜋/2+2𝑘𝜋−𝜋/2+2𝑘𝜋}𝛾sin²𝑡cos𝑡𝐼₂−−cos𝑡−sin𝑡𝑡sin𝑡𝑡×−cos𝑡−cos𝑡sin𝑡𝑡−sin𝑡𝑡=−cos𝑡𝑑𝑡_{𝜋/2+2𝑘𝜋−𝜋/2+2𝑘𝜋}𝛾sin²𝑡cos𝑡−cos²𝑡−sin²𝑡𝑡²𝑑𝑡𝐼₂≥^{𝜋/2+2𝑘𝜋}_{−𝜋/2+2𝑘𝜋}𝛾sin²𝑡cos𝑡−cos²𝑡−sin²𝑡𝑑𝑡𝐼₂=^𝜋/2_−𝜋/2sin²𝑡𝛾cos𝑡−1𝑑𝑡𝐼₂1>0for𝛾>2. (29) So we obtain that the Hamiltonian system is oscillatory for 𝛾>1/2.

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).