Oscillation Criteria Based on a New Weighted Function for Linear Matrix Hamiltonian Systems

Abstract

By employing a generalized Riccati technique and an integral averaging technique, some new oscillation criteria are established for the second-order matrix differential system 𝑈^′=𝐴(𝑥)𝑈+𝐵(𝑡)𝑉, 𝑉^′=𝐶(𝑥)𝑈−𝐴^∗(𝑡)𝑉, where 𝐴(𝑡), 𝐵(𝑡), and 𝐶(𝑡) are (𝑛×𝑛)-matrices, and 𝐵, 𝐶 are Hermitian. These results are sharper than some previous results.

1. Introduction

In this paper, we are concerned with the oscillatory behavior of the linear matrix Hamiltonian system of the form 𝑈^′(𝑥)(𝑡)𝑉=𝐴𝑈+𝐵𝑉,_′=𝐶(𝑥)𝑈−𝐴^∗(𝑡)𝑉,𝑡≥𝑡₀, (1.1) where 𝐴(𝑡), 𝐵(𝑡), and 𝐶(𝑡) are (𝑛×𝑛)-matrices and 𝐵, 𝐶 are Hermitian, that is, 𝐵^∗(𝑡)=𝐵(𝑡), 𝐶^∗(𝑡)=𝐶(𝑡). For any matrix 𝐴, the transpose of 𝐴 is denoted by 𝐴^∗.

For any real symmetric matrixes 𝑃, 𝑄, 𝑅, we write 𝑃≥𝑄 meaning that 𝑃−𝑄≥0; that is, 𝑃−𝑄 is positive semidefinite and 𝑃>𝑄 meaning that 𝑃−𝑄>0; that is, 𝑃−𝑄 is positive definite.

Definition 1.1. A solution (𝑈(𝑡),𝑉(𝑡)) of (1.1) is called nontrivial if det𝑈(𝑡)≠0 for at least one 𝑡≥𝑡₀.

Definition 1.2. A nontrivial solution (𝑈(𝑡),𝑉(𝑡)) of (1.1) is called prepared if 𝑈^∗(𝑡)𝑉(𝑡)−𝑉^∗(𝑡)𝑈(𝑇)=0 for every 𝑡≥𝑡₀.

Definition 1.3. System (1.1) is called oscillatory on [𝑡₀,∞) if there is a nontrivial prepared solution (𝑈(𝑡),𝑉(𝑡)) of (1.1) having the property that det𝑈(𝑡) vanishes on [𝑇,∞) for every 𝑇>𝑡₀. Otherwise, it is called nonoscillatory.

Note 1. It follows from [1, Theorem  8.1, page 303] that if the system (1.1) is oscillatory on [𝑡₀,∞), then every nontrivial prepared solution (𝑈(𝑡),𝑉(𝑡)) of (1.1) has the property that det𝑈(𝑡) vanishes on [𝑇,∞) for every 𝑇>𝑡₀.

The oscillation problem for system (1.1) and its various particular cases such as the second-order matrix differential systems[𝑌(𝑡)]^′′+𝑄(𝑡)𝑌(𝑡)=0,𝑡∈𝑡₀,[𝑃(𝑡)𝑌(𝑡,∞)]^′′+𝑄(𝑡)𝑌(𝑡)=0,𝑡∈𝑡₀,∞, (1.2)(1.3) has been studied extensively in recent years, for example, see [1–23]. Some of the most important conditions that guarantee that system (1.2) is oscillatory are as follows: lim_𝑡→∞𝜆₁{∫^𝑡_𝑡0𝑄(𝑠)𝑑𝑠}=∞ (see [4, 6]),lim_𝑡→∞inf(1/𝑡)∫^𝑡_𝑡0∫^𝑠_𝑡0tr𝑄(𝜏)𝑑𝜏𝑑𝑠>−∞ andlim_𝑡→∞sup(1/𝑡)∫^𝑡_𝑡0𝜆₁[∫^𝑠_𝑡0𝑄(𝜏)𝑑𝜏]𝑑𝑠=∞ orlim_𝑡→∞sup(1/𝑡)∫^𝑡_𝑡0{𝜆₁[∫^𝑠_𝑡0𝑄(𝜏)𝑑𝜏]}²𝑑𝑠=∞ (see [5]),lim_𝑡→∞sup(1/𝑡^𝑚−1)𝜆₁[∫^𝑡_𝑡0(𝑡−𝑠)^𝑚−1𝑄(𝑠)𝑑𝑠]𝑑𝑠=∞, 𝑚>2 is an integer (see [2]).

We particularly mention the other results of Erbe et al. [2] who proved the following theorem.

Erbe, Kong, and Ruan's Theorem
Let 𝐻(𝑡,𝑠) and ℎ(𝑡,𝑠) be continuous on 𝐷={𝑓(𝑡,𝑠)∶𝑡≥𝑠≥𝑡₀} such that 𝐻(𝑡,𝑡)=0 for 𝑡≥𝑡₀ and 𝐻(𝑡,𝑠)>0 for 𝑡>𝑠≥𝑡₀. We assume further that the partial derivative (𝜕/𝜕𝑠)𝐻(𝑡,𝑠)=𝐻_𝑠(𝑡,𝑠) is nonpositive and continuous for 𝑡≥𝑠≥𝑡₀ and ℎ(𝑡,𝑠) is defined by 𝐻_𝑠(𝑡,𝑠)=−ℎ(𝑡,𝑠)[𝐻(𝑡,𝑠)]^1/2,(𝑡,𝑠)∈𝐷. (1.4) Finally, we assume that lim_𝑡→∞1sup𝐻𝑡,𝑡₀𝜆₁^𝑡_𝑡0𝐻()𝑄(𝑠)−1𝑡,𝑠4ℎ²(𝑡,𝑠)𝐼𝑑𝑠=∞, (1.5) where 𝜆₁[𝐴]≥𝜆₂[𝐴]≥⋯≥𝜆_𝑛[𝐴] denotes the usual ordering of the eigenvalues of the symmetric matrix 𝐴; 𝐼 is the 𝑛×𝑛 identity matrix. Then system (1.2) is oscillatory.

And, later, Meng et al. [3] gave the following oscillation criteria.

Meng, Wang, and Zheng's Theorem
Let 𝐻(𝑡,𝑠) and ℎ(𝑡,𝑠) be continuous on 𝐷={(𝑡,𝑠)∶𝑡≥𝑠≥𝑡₀} such that 𝐻(𝑡,𝑡)=0 for 𝑡≥𝑡₀ and 𝐻(𝑡,𝑠)>0 for 𝑡>𝑠≥𝑡₀. We assume further that the partial derivative (𝜕/𝜕𝑠)𝐻(𝑡,𝑠)=𝐻_𝑠(𝑡,𝑠) is nonpositive and continuous for 𝑡≥𝑠≥𝑡₀ and ℎ(𝑡,𝑠) is defined by 𝐻_𝑠(𝑡,𝑠)=−ℎ(𝑡,𝑠)[𝐻(𝑡,𝑠)]^1/2,(𝑡,𝑠)∈𝐷. (1.6) If there exists a function 𝑓∈𝐶¹[𝑡₀,∞) such that lim_𝑡→∞1sup𝐻𝑡,𝑡₀𝜆₁^𝑡_𝑡0𝐻()𝑅(𝑠)−1𝑡,𝑠4𝑎(𝑠)ℎ²(𝑡,𝑠)𝐼𝑑𝑠=∞, (1.7) where 𝑎(𝑡)=exp{−2∫𝑡𝑓(𝑠)𝑑𝑠}, 𝑅(𝑡)=𝑎(𝑡){𝑄(𝑡)+𝑓²(𝑡)𝐼−𝑓′(𝑡)𝐼}. Then system (1.2) is oscillatory.

However, all these results are given in the form of lim_𝑡→∞sup𝜆₁[⋅]=+∞. In this paper, using the generalized Riccati technique and the integral averaging technique, we establish some new oscillation criteria which are different from most known ones in the sense that they are based on a new weighted function ℏ(𝑡,𝑠,𝑙) and which are presented in the form of lim_𝑡→∞sup𝜆₁[⋅]> const. Our results are presented in the form of a high degree of generality. Although the conditions in our main results (Theorem 2.1) seem to be more complicated compared to the known ones, with appropriate choices of the functions ℏ, 𝑓, we derive a number of oscillation criteria (see also (2.2)), which extend, improve, and unify a number of existing results and handle the cases not covered by known criteria. In particular, this can be seen by the examples given at the end of this paper.

2. Main Results

In the last literature, most oscillation results involve a function 𝐻=𝐻(𝑡,𝑠)∈𝐶(𝐷,𝑅+), where 𝐷={(𝑡,𝑠)∶𝑡₀≤𝑠≤𝑡<∞}, which satisfies 𝐻(𝑡,𝑡)=0, 𝐻(𝑡,𝑠)>0 for 𝑡>𝑠 and has partial derivative 𝜕𝐻/𝜕𝑠 on 𝐷 such that 𝜕𝐻(𝐻(𝜕𝑠=−ℎ𝑡,𝑠)[𝑡,𝑠)]^1/2, (2.1) where ℎ is locally integrable with respect to 𝑠 in 𝐷.

In this paper, let a function ℏ=ℏ(𝑡,𝑠,𝑙) be continuous on 𝐷={(𝑡,𝑠,𝑙)∶𝑡₀≤𝑙≤𝑠≤𝑡<+∞}, which satisfies ℏ(𝑡,𝑡,𝑙)=0, ℏ(𝑡,𝑠,𝑙)>0 for 𝑙≤𝑠<𝑡 and has the partial derivative 𝜕ℏ/𝜕𝑠 on 𝐷 such that 𝜕ℏ/𝜕𝑠 is locally integrable with respect to 𝑠 in 𝐷, and we call the two positive numbers 𝛾 and 𝛿 admissible [22] if they satisfy the condition 𝛾𝛿>1.

Theorem 2.1. If there exist a function 𝑓∈𝐶¹[𝑡₀,∞) and two admissible numbers 𝛾, 𝛿 such that lim_𝑡→∞1supℏ²𝑡,𝑡₀,𝑡₀𝜆₁^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀Ψ(𝑠)+𝛾𝛿𝑃𝑡,𝑠,𝑡₀𝑑𝑠=∞, (2.2) where Ψ(𝑠)=𝑏(𝑠)[−𝐶−𝑓(𝐴+𝐴^∗)+𝑓²𝐵−𝑓^′𝐼](𝑠), 𝐼 is the 𝑛×𝑛 identity matrix, 𝑏(𝑠)=exp(−2∫^𝑠_𝑥0𝑓(𝜍)𝑑𝜍), and 𝑃𝑡,𝑠,𝑡₀=𝑏(𝑠)ℏ²𝑡,𝑠,𝑡₀𝑓𝐴+𝐴^∗−𝐴^∗𝐵⁻¹𝐴𝑠)(𝑠)ℏ(−𝑏𝑡,𝑠,𝑡₀ℏ^′_𝑠𝑡,𝑠,𝑡₀−𝑓(𝑠)ℏ𝑡,𝑠,𝑡₀×𝐴^∗𝐵⁻¹+𝐵⁻¹𝐴𝑠)(𝑠ℏ(−𝑏)^′_𝑠𝑡,𝑠,𝑡₀−𝑓(𝑠)ℏ𝑡,𝑠,𝑡₀𝐵^−1/2(𝑠)−𝑓(𝑠)ℏ𝑡,𝑠,𝑡₀𝐵^1/2(𝑠)², (2.3) then system (1.1) is oscillatory.

Proof. Suppose to the contrary that system (1.1) is nonoscillatory. Then there exists a nontrivial prepared solution (𝑈(𝑡),𝑉(𝑡)) of (1.1) such that 𝑈(𝑡) is nonsingular on [𝑇,∞) for some 𝑇>𝑡₀. Without loss of generality, we may assume that det𝑈(𝑡)≠0 for 𝑡≥𝑡₀. Define 𝑊(𝑡)=𝑏(𝑡)𝑉(𝑡)𝑈⁻¹(𝑡)+𝑓(𝑡)𝐼,𝑡≥𝑡₀. (2.4) Then 𝑊(𝑡) is well defined, Hermitian, and it satisfies the Riccati equation 𝑊^′+𝑊𝐴+𝐴^∗1𝑊+𝑏𝑊𝐵𝑊−𝑓[𝑊𝐵+𝐵𝑊−2𝑊]+Ψ(𝑡)=0 (2.5) on [𝑡₀,∞). Multiplying (2.5), with 𝑡 replaced by 𝑠, by ℏ²(𝑡,𝑠,𝑡₀), integrating from 𝑡₀ to 𝑡, and picking two admissible numbers 𝛾 and 𝛿, we obtain ^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀Ψ(𝑠)𝑑𝑠=−^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀𝑊′(𝑠)𝑑𝑠−^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀𝑏(𝑠)[𝑠)−𝑊𝐵𝑊](𝑑𝑠_𝑡𝑡0ℏ²𝑡,𝑠,𝑡₀𝑊𝐴+𝐴^∗𝑊−𝑓(𝑊𝐵+𝐵𝑊−2𝑊)(𝑠)𝑑𝑠=ℏ²𝑡,𝑡₀,𝑡₀𝑊𝑡₀−^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀𝑏(𝑠)[𝑠)−𝑊𝐵𝑊](𝑑𝑠_𝑡𝑡0ℏ²𝑡,𝑠,𝑡₀𝑊𝐴+𝐴^∗𝑊−𝑓(𝑊𝐵+𝐵𝑊)(𝑠)𝑑𝑠+2^𝑡_𝑡0ℏ𝑡,𝑠,𝑡₀ℏ^′_𝑠𝑡,𝑠,𝑡₀−𝑓(𝑠)ℏ𝑡,𝑠,𝑡₀𝑊(𝑠)𝑑𝑠=ℏ²𝑡,𝑡₀,𝑡₀𝑊𝑡₀−1𝛾^𝑡_𝑡0𝑄^∗𝑄𝑡,𝑠,𝑡₀𝑑𝑠−𝛾𝛿^𝑡_𝑡0𝑃𝑡,𝑠,𝑡₀−𝑑𝑠𝛾𝛿−1𝛾𝛿^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀𝑏(𝑠)(𝑅𝑊)^∗(𝑅𝑊)(𝑠)𝑑𝑠, (2.6) where 𝑅(𝑡)=𝐵(𝑡) and 𝑄𝑡,𝑠,𝑡₀=ℏ𝑡,𝑠,𝑡₀(𝑠)(𝑠)𝛿𝑏𝑅𝑊)(−𝛾𝛿𝑏(𝑠)ℏ𝑡,𝑠,𝑡₀𝑓𝑅−𝑅⁻¹𝐴(𝑠)+𝛾𝛿𝑏(𝑠)ℏ^′_𝑠𝑡,𝑠,𝑡₀−𝑓(𝑠)ℏ𝑡,𝑠,𝑡₀𝑅⁻¹(𝑠). (2.7) Then ^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀Ψ(𝑠)+𝛾𝛿𝑃𝑡,𝑠,𝑡₀𝑑𝑠=ℏ²𝑡,𝑡₀,𝑡₀𝑊𝑡₀−1𝛾^𝑡_𝑡0𝑄^∗𝑄𝑡,𝑠,𝑡₀−𝑑𝑠𝛾𝛿−1𝛾𝛿^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀𝑏(𝑠)(𝑅𝑊)^∗(𝑅𝑊)(𝑠)𝑑𝑠≤ℏ²𝑡,𝑡₀,𝑡₀𝑊𝑡₀. (2.8) This implies that 𝜆₁^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀Ψ(𝑠)+𝛾𝛿𝑃𝑡,𝑠,𝑡₀𝑑𝑠≤ℏ²𝑡,𝑡₀,𝑡₀𝜆₁𝑊𝑡₀, (2.9) and then 1ℏ²𝑡,𝑡₀,𝑡₀𝜆₁^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀Ψ(𝑠)+𝛾𝛿𝑃𝑡,𝑠,𝑡₀𝑑𝑠≤𝜆₁𝑊𝑡₀. (2.10) Taking the upper limit in both sides of (2.10) as 𝑡→∞, the right-hand side is always bounded, which contradicts condition (2.2). This completes the proof of Theorem 2.1.

By applying the matrix theory [8, 21], we have the following theorem from Theorem 2.1.

Theorem 2.2. If there exist a function 𝑓∈𝐶¹[𝑡₀,∞) and two admissible numbers 𝛾, 𝛿 such that lim_𝑡→∞1supℏ²𝑡,𝑡₀,𝑡₀^𝑡𝑡₀ℏ²𝑡,𝑠,𝑡₀trΨ(s)+𝛾𝛿tr𝑃𝑡,𝑠,𝑡₀𝑑𝑠=∞, (2.11) where Ψ(𝑠), 𝑏(𝑠), and 𝑃(𝑡,𝑠,𝑡₀) are as in Theorem 2.1, then system (1.1) is oscillatory.

By [8], the trace tr∶𝑆→𝑅 is a positive linear functional on 𝑆, where the space 𝑆 is the linear space of all real symmetric 𝑛×𝑛 matrices. And noting that two admissible numbers 𝛾, 𝛿 satisfying 𝛾𝛿>1, then we have the following corollary from Theorem 2.2.

Corollary 2.3. If there exist a function 𝑓∈𝐶¹[𝑡₀,∞) and two admissible numbers 𝛾, 𝛿 such that lim_𝑡→∞1supℏ²𝑡,𝑡₀,𝑡₀^𝑡𝑡₀ℏ²𝑡,𝑠,𝑡₀trΨ(𝑠)+tr𝑃𝑡,𝑠,𝑡₀𝑑𝑠=∞, (2.12) where Ψ(𝑠), 𝑏(𝑠), and 𝑃(𝑡,𝑠,𝑡₀) are as in Theorem 2.1, then system (1.1) is oscillatory.

Proof. By virtue of a simple property of limits lim_𝑡→∞1supℏ²𝑡,𝑡₀,𝑡₀^𝑡𝑡₀ℏ²𝑡,𝑠,𝑡₀trΨ(𝑠)+𝛾𝛿tr𝑃𝑡,𝑠,𝑡₀𝑑𝑠>lim_𝑡→∞1supℏ²𝑡,𝑡₀,𝑡₀^𝑡𝑡₀ℏ²𝑡,𝑠,𝑡₀trΨ(𝑠)+tr𝑃𝑡,𝑠,𝑡₀𝑑𝑠 (2.13) and (2.12), the conclusion follows from Theorem 2.2.

If we choose ℏ(𝑡,𝑠,𝑡₀)=𝐻(𝑡,𝑠)/𝐻(𝑡,𝑡₀) in Theorem 2.1, then ℏ𝑡,𝑡₀,𝑡₀=𝐻𝑡,𝑡₀𝐻𝑡,𝑡₀=1, (2.14) we have the following.

Corollary 2.4. If there exist a function 𝑓∈𝐶¹[𝑡₀,∞) and two admissible numbers 𝛾, 𝛿 such that lim_𝑡→∞1sup𝐻𝑡,𝑡₀𝜆₁^𝑡_𝑡0𝐻(𝑡,𝑠)Ψ₁(𝑠)+𝛾𝛿𝑃₁(𝑡,𝑠)𝑑𝑠=∞, (2.15) where 𝐻(𝑡,𝑠) are as in Erbe, Kong, and Ruan's Theorem, Ψ₁(𝑠)=𝑏(𝑠)[−𝐶−𝛾𝛿𝐴^∗𝐵⁻¹𝐴+𝑓²𝐵−𝑓′𝐼+𝛾𝛿𝑓(𝐴^∗𝐵⁻¹+𝐵⁻¹𝐴)](𝑠), 𝐼 is the 𝑛×𝑛 identity matrix, 𝑏(𝑠)=exp(−2∫^𝑠_𝑥0𝑓(𝜍)𝑑𝜍), and 𝑃₁()=𝑏(𝑠)ℎ()𝑡,𝑠𝑡,𝑠𝐻(𝑡,𝑠)2𝐴^∗𝐵⁻¹+𝐵⁻¹𝐴𝑠)(𝑠ℎ()(−𝑏)𝑡,𝑠2+𝑓(𝑠)𝐻(𝑡,𝑠)𝐵^−1/2(𝑠)+𝑓(𝑠)𝐻(𝑡,𝑠)𝐵^1/2(𝑠)², (2.16) then system (1.1) is oscillatory.

Remark 2.5. In the last literature [1–4, 12, 15, 23], most oscillation results were given in the form of lim_𝑡→∞sup(1/(𝐻(𝑡,𝑡₀)))𝜆₁[⋅]=+∞. Obviously, Theorem 2.1 extends and improves a number of existing results and handles the cases not covered by known criteria, which can be seen from Corollary 2.4.

If we choose 𝑓(𝑡)=0 and let ℏ(𝑡,𝑠,𝑟)=(𝑡−𝑠)^𝛼/(𝑡−𝑟)^𝛽 for 𝛼,𝛽>1/2 in Theorem 2.1, then we have the following.

Corollary 2.6. If there exist two real numbers 𝛼,𝛽>1/2 and two admissible numbers 𝛾, 𝛿 such that lim_𝑡→∞1sup𝑡^𝛼𝜆₁^𝑡_𝑡0(𝑡−𝑠)^𝛼𝛼()𝛾𝛿𝑡−𝑠^𝛼−12𝐴^∗𝐵⁻¹+𝐵⁻¹𝐴−𝐴^∗𝐵⁻¹𝐴−𝛼(𝑡−𝑠)^2(𝛼−1)4𝐵⁻¹−𝛾𝛿𝐶𝑑𝑠=∞, (2.17) then system (1.1) is oscillatory.

If we choose appropriate 𝑓 in Theorem 2.1 such that 𝑏(𝑡)𝐵⁻¹(𝑡)≤𝐼 for 𝑡≥𝑡₀ and let ℏ(𝑡,𝑠,𝑟)=(𝑡−𝑠)^𝛼/(𝑡−𝑟) for 𝛼>2, then we have the following

Corollary 2.7. If there exist a function 𝑓∈𝐶¹[𝑡₀,∞) and two admissible numbers 𝛾, 𝛿 such that for some 𝛼>1/2 and for every 𝑟≥𝑡₀, lim_𝑡→∞1sup𝑡^2𝛼+1𝜆₁^𝑡_𝑟(𝑡−𝑠)²(𝑠−𝑟)^2𝛼Ψ(𝑠)+𝛾𝛿𝑃₁()>𝛼𝑡,𝑠,𝑟𝑑𝑠(2𝛼−1)(2𝛼+1), (2.18) where Ψ(𝑠), 𝑏(𝑠) are as in Theorem 2.1 and 𝑃₁(𝑡,𝑠,𝑟)=𝑏(𝑠)𝑓𝐴+𝐴^∗−𝐴^∗𝐵⁻¹𝐴𝑠)−𝑏(𝑠)(()[()𝑠𝐴𝑡−𝑠)(𝑠−𝑟𝑟+𝛼𝑡−𝛼+1]^∗𝐵⁻¹+𝐵⁻¹𝐴𝑠)(𝑠)𝑓(𝑠𝐴(+𝑏)^∗𝐵⁻¹+𝐵⁻¹𝐴−𝑓𝐵+2𝐼+𝐵⁻¹𝑠)+(𝑠)𝑓(𝑠)(2𝑏()[()𝑠]𝑡−𝑠)(𝑠−𝑟𝑟+𝛼𝑡−𝛼+1𝐼+𝐵⁻¹−𝑏(𝑠)(𝑡−𝑠)²(𝑠−𝑟)²𝐵⁻¹, (2.19) then system (1.1) is oscillatory.

Proof. Assume to the contrary that (1.1) is nonoscillatory. Then 𝑈(𝑡) is nonsingular for all sufficiently large 𝑡, say 𝑡≥𝑇≥𝑡₀. Similar to the proof of Theorem 2.1, for 𝑡≥𝑇≥𝑡₀, we have ^𝑡_𝑇ℏ²(𝑡,𝑠,𝑇)Ψ(𝑠)+𝛾𝛿𝑃₁(𝑡,𝑠,𝑇)𝑑𝑠≤𝛾𝛿^𝑡_𝑇𝑏(𝑠)𝐵⁻¹(𝑠)ℏ^′_𝑠(𝑡,𝑠,𝑇)²𝑑𝑠≤𝛾𝛿^𝑡_𝑇(𝑠−𝑇)^2(𝛼−1)[𝑇+𝛼𝑡−(𝛼+1)𝑠]²𝐼. (2.20) This implies that 𝜆₁^𝑡_𝑇ℏ²(𝑡,𝑠,𝑇)Ψ(𝑠)+𝛾𝛿𝑃₁(𝑡,𝑠,𝑇)𝑑𝑠≤^𝑡_𝑇(𝑠−𝑇)^2(𝛼−1)[𝑇+𝛼𝑡−(𝛼+1)𝑠]²=𝛼𝑑𝑠()()2𝛼−1)(2𝛼+1𝑡−𝑇^2(𝛼+1). (2.21) Then lim_𝑡→∞1sup𝑡^2𝛼+1𝜆₁^𝑡_𝑇ℏ²(𝑡,𝑠,𝑇)Ψ(𝑠)+𝛾𝛿𝑃₁()≤𝛼𝑡,𝑠,𝑇𝑑𝑠(2𝛼−1)(2𝛼+1), (2.22) which contradicts assumption (2.18). This completes the proof of Corollary 2.7.

When 𝐴(𝑡)≡0, 𝐵⁻¹(𝑡)=𝑃(𝑡) and −𝐶(𝑡)=𝑄(𝑡) for 𝑡≥𝑡₀, then system (1.1) reduces to system (1.3).

As an immediate result of Theorem 2.1, we have the following theorem.

Theorem 2.8. If there exist a function 𝑓∈𝐶¹[𝑡₀,∞) and two admissible numbers 𝛾, 𝛿 such that lim_𝑡→∞1supℏ²𝑡,𝑡₀,𝑡₀𝜆₁^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀Ψ₁(𝑠)+𝛾𝛿𝑃₂𝑡,𝑠,𝑡₀𝑑𝑠=∞, (2.23) where Ψ₁(𝑠)=𝑏(𝑠)[𝑄(𝑡)+𝑓²𝑃⁻¹(𝑡)−𝑓′𝐼](𝑠), 𝐼 is the 𝑛×𝑛 identity matrix, 𝑏(𝑠)=exp(−2∫^𝑠_𝑥0𝑓(𝜍)𝑑𝜍), and 𝑃₂𝑡,𝑠,𝑡₀=−𝑏(𝑠)ℏ^′_𝑠𝑡,𝑠,𝑡₀−𝑓(𝑠)ℏ𝑡,𝑠,𝑡₀𝑃^1/2(𝑠)−𝑓(𝑠)ℏ𝑡,𝑠,𝑡₀𝑃^−1/2(𝑠)², (2.24) then system (1.3) is oscillatory.

By applying the matrix theory [8, 21], we have the following theorem from Theorem 2.8.

Theorem 2.9. If there exist a function 𝑓∈𝐶¹[𝑡₀,∞) and two admissible numbers 𝛾,𝛿 such that lim_𝑡→∞sup𝜆₁^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀trΨ₁(𝑠)+𝛾𝛿tr𝑃₂𝑡,𝑠,𝑡₀𝑑𝑠=∞, (2.25) where Ψ₁(𝑠), 𝑏(𝑠), and 𝑃₂(𝑡,𝑠,𝑡₀) are as in Theorem 2.8, then system (1.3) is oscillatory.

By [8], the trace tr∶𝑆→𝑅 is a positive linear functional on 𝑆, where the space 𝑆 is the linear space of all real symmetric 𝑛×𝑛 matrices. And noting that two admissible numbers 𝛾, 𝛿 satisfying 𝛾𝛿>1, then we have the following corollary from Theorem 2.9.

Corollary 2.10. If there exist a function 𝑓∈𝐶¹[𝑡₀,∞) and two admissible numbers 𝛾, 𝛿 such that lim_𝑡→∞sup𝜆₁^𝑡_𝑡0ℏ²𝑡,𝑠,𝑡₀trΨ₁(𝑠)+tr𝑃₂𝑡,𝑠,𝑡₀𝑑𝑠=∞, (2.26) where Ψ₁(𝑠), 𝑏(𝑠), and 𝑃₂(𝑡,𝑠,𝑡₀) are as in Theorem 2.8, then system (1.3) is oscillatory.

By Corollary 2.7 and (1.3), we easily get the following theorem:

Theorem 2.11. If there exist a function 𝑓∈𝐶¹[𝑡₀,∞) and two admissible numbers 𝛾, 𝛿 such that for some 𝛼>1/2 and for every 𝑟≥𝑡₀, lim_𝑡→∞1sup𝑡^2𝛼+1𝜆₁^𝑡_𝑟(𝑡−𝑠)²(𝑠−𝑟)^2𝛼Ψ₁(𝑠)+𝛾𝛿𝑃₃()>𝛼𝑡,𝑠,𝑟𝑑𝑠(2𝛼−1)(2𝛼+1), (2.27) where Ψ₁(𝑠), 𝑏(𝑠) are as in Theorem 2.8 and 𝑃₃(𝑡,𝑠,𝑟)=−𝑏(𝑠)𝑓²(𝑠)𝑃+2𝐼+𝑃⁻¹(𝑠)−𝑏(𝑠)(𝑡−𝑠)²(𝑠−𝑟)²𝑃(𝑠)+2𝑏(𝑠)𝑓(𝑠)()[()𝑠𝑠),𝑡−𝑠)(𝑠−𝑟𝑟+𝛼𝑡−𝛼+1](𝐼+𝑃)( (2.28) then system (1.3) is oscillatory.

3. Examples

Example 3.1. Consider the Euler differential system 𝑌^′′𝑛+diag𝑡²,𝑚𝑡²𝑌=0,𝑡≥1,𝑚≥𝑛>0. (3.1) If we choose 𝑓(𝑡)=0, then 𝑎(𝑡)=1, Ψ₁(𝑡)=diag(𝑛/𝑡²,𝑚/𝑡²) and 𝑃₃(𝑡,𝑠,𝑟)=(1/((𝑡−𝑠)²(𝑠−𝑟)²))𝐼. Note that for each 𝑟≥1, lim_𝑡→∞1𝑡^2𝛼+1𝑡_𝑟(𝑡−𝑠)²(𝑠−𝑟)^2𝛼𝑚𝑡²−𝛾𝛿(𝑡−𝑠)²(𝑠−𝑟)²𝑑𝑠=lim_𝑡→∞1𝑡^2𝛼+1𝑡_𝑟𝑚(𝑡−𝑠)²(𝑠−𝑟)^2𝛼𝑡²𝑑𝑠−lim_𝑡→∞𝛾𝛿𝑡^2𝛼+1𝑡_𝑟(𝑠−𝑟)^2𝛼−2=𝑚𝑑𝑠𝛼().2𝛼−1)(2𝛼+1 (3.2) Obviously, for any 𝑚>1/4, there exists 𝛼>1/2 such that 𝑚𝛼()>𝛼2𝛼−1)(2𝛼+1(2𝛼−1)(2𝛼+1). (3.3) This means that (2.25) holds. By Theorem 2.11, we find that system (3.1) is oscillatory for 𝑚>1/4.

Remark 3.2. As pointed out in [3], the above-mentioned criteria (1.5) of Erbe, Kong, and Ruan cannot be applied to the Euler differential system (3.1), for lim_𝑡→∞1sup𝐻()𝜆𝑡,1^1𝑡₁𝐻()𝑄(𝑠)−1𝑡,𝑠4ℎ²(𝑡,𝑠)𝐼𝑑𝑠≤lim_𝑡→∞^𝑡₁𝑚𝑠²𝑑𝑠=𝑚<∞. (3.4) Though the above-mentioned criteria (1.7) of Meng, Wang, and Zheng's Theorem can be applied to the Euler differential system, our results are sharper than theirs, which can be seen from Example 3.1.

Remark 3.3. It is interesting for the fact that If we choose 𝑓(𝑡)=0, ℏ(𝑡,𝑠,𝑇)=𝐻(𝑡,𝑠)/𝐻(𝑡,𝑇), then for differential system (1.2), we have lim_𝑡→∞sup𝜆₁^𝑡₁ℏ²(𝑡,𝑠,1)Ψ₁(𝑠)+𝑃₂(𝑡,𝑠,1)𝑑𝑠=lim_𝑡→∞1sup𝐻()𝜆𝑡,1^1𝑡₁𝐻()𝑄(𝑠)−1𝑡,𝑠4ℎ²(𝑡,𝑠)𝐼𝑑𝑠, (3.5) where 𝐻(𝑡,𝑠) are as in Erbe, Kong, and Ruan's Theorem. Obviously, Theorem 2.8 extends and improves a number of existing results and handles the cases not covered by known criteria.

Example 3.4. Consider the 4-dimensional system (1.1) where 𝐴(𝑡)≡0,𝐵(𝑡)=(𝑡+1)²𝐼₂(𝑡)𝜌,𝐶=−𝑡²00𝜎2𝑡², (3.6) and where 𝜌≥𝜎>0 and 𝑡≥1. If we let 𝑓(𝑡)=0, then 𝑏(𝑡)=1 and 𝑏(𝑡)𝐵⁻¹(𝑡)≤𝐼₂ for 𝑡≥1. Thus, we have Ψ(𝑠)=𝜌𝑡²00𝜎2𝑡²,𝑃₁()1𝑡,𝑠,𝑟=−(𝑡−𝑠)²(𝑠−𝑟)²(𝑠+1)²𝐼₂. (3.7) Thus, if we choose two admissible numbers 𝛾, 𝛿 such that 𝛾𝛿=3/2, then for some 𝛼>1/2 and for every 𝑟≥𝑡₀, lim_𝑡→∞1𝑡^2𝛼+1𝑡_𝑟(𝑡−𝑠)²(𝑠−𝑟)^2𝛼𝜌𝑡²−32(𝑡−𝑠)²(𝑠−𝑟)²(𝑠+1)²=𝜌𝑑𝑠𝛼(2𝛼−1)(2𝛼+1). (3.8) By Corollary 2.6, we find that system (3.1) is oscillatory for 𝜌>1/4.

Acknowledgment

This wrok was supported financially by the NNSF of China (10801088).