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

Authors: Yingxin Guo; Junchang Wang
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

By employing a generalized Riccati technique and an integral averaging technique, some new oscillation criteria are established for the second-order matrix differential systemU′=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.


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## Body

## 1. Introduction

In this paper, we are concerned with the oscillatory behavior of the linear matrix Hamiltonian system of the form(1.1)U′=A(x)U+B(t)V,V′=C(x)U-A*(t)V,t≥t0,
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 matrixesP, 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(1.2)[Y(t)]′′+Q(t)Y(t)=0,t∈[t0,∞),(1.3)[P(t)Y(t)]′′+Q(t)Y(t)=0,t∈[t0,∞),
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: limt→∞λ1{∫t0tQ(s)ds}=∞      (see [4, 6]),limt→∞inf(1/t)∫t0t∫t0strQ(τ)dτds>-∞ andlimt→∞sup(1/t)∫t0tλ1[∫t0sQ(τ)dτ]ds=∞ orlimt→∞sup(1/t)∫t0t{λ1[∫t0sQ(τ)dτ]}2ds=∞ (see [5]),limt→∞sup(1/tm-1)λ1[∫t0t(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
LetH(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)=Hs(t,s) is nonpositive and continuous for t≥s≥t0 and h(t,s) is defined by
(1.4)Hs(t,s)=-h(t,s)[H(t,s)]1/2,(t,s)∈D.
Finally, we assume that
(1.5)lim⁡t→∞sup⁡1H(t,t0)λ1[∫t0t(H(t,s)Q(s)-14h2(t,s)I)ds]=∞,
where λ1[A]≥λ2[A]≥⋯≥λn[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
LetH(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)=Hs(t,s) is nonpositive and continuous for t≥s≥t0 and h(t,s) is defined by
(1.6)Hs(t,s)=-h(t,s)[H(t,s)]1/2,(t,s)∈D.
If there exists a function f∈C1[t0,∞) such that
(1.7)lim⁡t→∞sup⁡1H(t,t0)λ1[∫t0t(H(t,s)R(s)-14a(s)h2(t,s)I)ds]=∞,
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 oflimt→∞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 limt→∞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 functionH=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 (2.1)∂H∂s=-h(t,s)[H(t,s)]1/2,
where h is locally integrable with respect to s in D.In this paper, let a functionℏ=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 functionf∈C1[t0,∞) and two admissible numbers γ, δ such that
(2.2)lim⁡t→∞sup⁡1ℏ2(t,t0,t0)λ1[∫t0t(ℏ2(t,s,t0)Ψ(s)+γδP(t,s,t0))ds]=∞,
where Ψ(s)=b(s)[-C-f(A+A*)+f2B-f′I](s), I is the n×n identity matrix, b(s)=exp(-2∫x0sf(ς)dς), and
(2.3)P(t,s,t0)=b(s)ℏ2(t,s,t0)[f(A+A*)-A*B-1A](s)-b(s)ℏ(t,s,t0)[ℏs′(t,s,t0)-f(s)ℏ(t,s,t0)]×[A*B-1+B-1A](s)-b(s)[(ℏs′(t,s,t0)-f(s)ℏ(t,s,t0))B-1/2(s)-f(s)ℏ(t,s,t0)B1/2(s)]2,
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
(2.4)W(t)=b(t)[V(t)U-1(t)+f(t)I],t≥t0.
Then W(t) is well defined, Hermitian, and it satisfies the Riccati equation
(2.5){W′+WA+A*W+1bWBW-f[WB+BW-2W]+Ψ}(t)=0
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
(2.6)∫t0tℏ2(t,s,t0)Ψ(s)ds=-∫t0tℏ2(t,s,t0)W′(s)ds-∫t0tℏ2(t,s,t0)b(s)[WBW](s)ds-∫t0tℏ2(t,s,t0)[WA+A*W-f(WB+BW-2W)](s)ds=ℏ2(t,t0,t0)W(t0)-∫t0tℏ2(t,s,t0)b(s)[WBW](s)ds-∫t0tℏ2(t,s,t0)[WA+A*W-f(WB+BW)](s)ds+2∫t0tℏ(t,s,t0)[ℏs′(t,s,t0)-f(s)ℏ(t,s,t0)]W(s)ds=ℏ2(t,t0,t0)W(t0)-1γ∫t0t[(Q*Q)(t,s,t0)]ds-γδ∫t0t[P(t,s,t0)]ds-γδ-1γδ∫t0tℏ2(t,s,t0)b(s)[(RW)*(RW)](s)ds,
where R(t)=B(t) and
(2.7)Q(t,s,t0)=ℏ(t,s,t0)δb(s)(RW)(s)-γ[δb(s)ℏ(t,s,t0)](fR-R-1A)(s)+γδb(s)[ℏs′(t,s,t0)-f(s)ℏ(t,s,t0)]R-1(s).
Then
(2.8)∫t0t(ℏ2(t,s,t0)Ψ(s)+γδP(t,s,t0))ds=ℏ2(t,t0,t0)W(t0)-1γ∫t0t[(Q*Q)(t,s,t0)]ds-γδ-1γδ∫t0tℏ2(t,s,t0)b(s)[(RW)*(RW)](s)ds≤ℏ2(t,t0,t0)W(t0).
This implies that
(2.9)λ1[∫t0t(ℏ2(t,s,t0)Ψ(s)+γδP(t,s,t0))ds]≤ℏ2(t,t0,t0)λ1(W(t0)),
and then
(2.10)1ℏ2(t,t0,t0)λ1[∫t0t(ℏ2(t,s,t0)Ψ(s)+γδP(t,s,t0))ds]≤λ1(W(t0)).
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 functionf∈C1[t0,∞) and two admissible numbers γ, δ such that
(2.11)lim⁡t→∞sup⁡1ℏ2(t,t0,t0)[∫t0t(ℏ2(t,s,t0)tr⁡Ψ(s)+γδtr⁡P(t,s,t0))ds]=∞,
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 functionf∈C1[t0,∞) and two admissible numbers γ, δ such that
(2.12)lim⁡t→∞sup⁡1ℏ2(t,t0,t0)[∫t0t(ℏ2(t,s,t0)tr⁡Ψ(s)+tr⁡P(t,s,t0))ds]=∞,
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(2.13)lim⁡t→∞sup⁡1ℏ2(t,t0,t0)[∫t0t(ℏ2(t,s,t0)tr⁡Ψ(s)+γδtr⁡P(t,s,t0))ds]>lim⁡t→∞sup⁡1ℏ2(t,t0,t0)[∫t0t(ℏ2(t,s,t0)tr⁡Ψ(s)+tr⁡P(t,s,t0))ds]
and (2.12), the conclusion follows from Theorem 2.2.If we chooseℏ(t,s,t0)=H(t,s)/H(t,t0) in Theorem 2.1, then (2.14)ℏ(t,t0,t0)=H(t,t0)H(t,t0)=1,
we have the following.Corollary 2.4.
If there exist a functionf∈C1[t0,∞) and two admissible numbers γ, δ such that
(2.15)lim⁡t→∞sup⁡1H(t,t0)λ1[∫t0t(H(t,s)Ψ1(s)+γδP1(t,s))ds]=∞,
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∫x0sf(ς)dς), and
(2.16)P1(t,s)=b(s)h(t,s)H(t,s)2[A*B-1+B-1A](s)-b(s)[(h(t,s)2+f(s)H(t,s))B-1/2(s)+f(s)H(t,s)B1/2(s)]2,
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 limt→∞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 choosef(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 numbersα,β>1/2 and two admissible numbers γ, δ such that
(2.17)lim⁡t→∞sup⁡1tαλ1{∫t0t(t-s)αγδ[α(t-s)α-12(A*B-1+B-1A)-A*B-1A-α(t-s)2(α-1)4B-1-γδC]ds}=∞,
then system (1.1) is oscillatory.If we choose appropriatef 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 followingCorollary 2.7.
If there exist a functionf∈C1[t0,∞) and two admissible numbers γ, δ such that for some α>1/2 and for every r≥t0,
(2.18)lim⁡t→∞sup⁡1t2α+1λ1[∫rt(t-s)2(s-r)2α(Ψ(s)+γδP1(t,s,r))ds]>α(2α-1)(2α+1),
where Ψ(s), b(s) are as in Theorem 2.1 and
(2.19)P1(t,s,r)=b(s)[f(A+A*)-A*B-1A](s)-b(s)(t-s)(s-r)[r+αt-(α+1)s][A*B-1+B-1A](s)+b(s)f(s)[A*B-1+B-1A-f(B+2I+B-1)](s)+2b(s)f(s)(t-s)(s-r)[r+αt-(α+1)s](I+B-1)-b(s)(t-s)2(s-r)2B-1,
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
(2.20)∫Tt(ℏ2(t,s,T)Ψ(s)+γδP1(t,s,T))ds≤γδ∫Ttb(s)B-1(s)[ℏs′(t,s,T)]2ds≤γδ∫Tt(s-T)2(α-1)[T+αt-(α+1)s]2I.
This implies that
(2.21)λ1[∫Tt(ℏ2(t,s,T)Ψ(s)+γδP1(t,s,T))ds]≤∫Tt(s-T)2(α-1)[T+αt-(α+1)s]2ds=α(2α-1)(2α+1)(t-T)2(α+1).
Then
(2.22)lim⁡t→∞sup⁡1t2α+1λ1[∫Tt(ℏ2(t,s,T)Ψ(s)+γδP1(t,s,T))ds]≤α(2α-1)(2α+1),
which contradicts assumption (2.18). This completes the proof of Corollary 2.7.WhenA(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 Theorem2.1, we have the following theorem.Theorem 2.8.
If there exist a functionf∈C1[t0,∞) and two admissible numbers γ, δ such that
(2.23)lim⁡t→∞sup⁡1ℏ2(t,t0,t0)λ1[∫t0t(ℏ2(t,s,t0)Ψ1(s)+γδP2(t,s,t0))ds]=∞,
where Ψ1(s)=b(s)[Q(t)+f2P-1(t)-f′I](s), I is the n×n identity matrix, b(s)=exp(-2∫x0sf(ς)dς), and
(2.24)P2(t,s,t0)=-b(s)[(ℏs′(t,s,t0)-f(s)ℏ(t,s,t0))P1/2(s)-f(s)ℏ(t,s,t0)P-1/2(s)]2,
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 functionf∈C1[t0,∞) and two admissible numbers γ,δ such that
(2.25)lim⁡t→∞sup⁡λ1[∫t0t(ℏ2(t,s,t0)tr⁡Ψ1(s)+γδtr⁡P2(t,s,t0))ds]=∞,
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 functionf∈C1[t0,∞) and two admissible numbers γ, δ such that
(2.26)lim⁡t→∞sup⁡λ1[∫t0t(ℏ2(t,s,t0)tr⁡Ψ1(s)+tr⁡P2(t,s,t0))ds]=∞,
where Ψ1(s), b(s), and P2(t,s,t0) are as in Theorem 2.8, then system (1.3) is oscillatory.By Corollary2.7 and (1.3), we easily get the following theorem:Theorem 2.11.
If there exist a functionf∈C1[t0,∞) and two admissible numbers γ, δ such that for some α>1/2 and for every r≥t0,
(2.27)lim⁡t→∞sup⁡1t2α+1λ1[∫rt(t-s)2(s-r)2α(Ψ1(s)+γδP3(t,s,r))ds]>α(2α-1)(2α+1),
where Ψ1(s), b(s) are as in Theorem 2.8 and
(2.28)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)(t-s)(s-r)[r+αt-(α+1)s](I+P)(s),
then system (1.3) is oscillatory.


## 3. Examples

Example 3.1.
Consider the Euler differential system(3.1)Y′′+diag⁡(nt2,mt2)Y=0,t≥1,m≥n>0.
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,
(3.2)lim⁡t→∞1t2α+1[∫rt(t-s)2(s-r)2α(mt2-γδ(t-s)2(s-r)2)ds]=lim⁡t→∞1t2α+1∫rtm(t-s)2(s-r)2αt2ds-lim⁡t→∞γδt2α+1∫rt(s-r)2α-2ds=mα(2α-1)(2α+1).
Obviously, for any m>1/4, there exists α>1/2 such that
(3.3)mα(2α-1)(2α+1)>α(2α-1)(2α+1).
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
(3.4)lim⁡t→∞sup⁡1H(t,1)λ1[∫1t(H(t,s)Q(s)-14h2(t,s)I)ds]≤lim⁡t→∞∫1tms2ds=m<∞.
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 choosef(t)=0, ℏ(t,s,T)=H(t,s)/H(t,T), then for differential system (1.2), we have
(3.5)lim⁡t→∞sup⁡λ1[∫1t(ℏ2(t,s,1)Ψ1(s)+P2(t,s,1))ds]=lim⁡t→∞sup⁡1H(t,1)λ1[∫1t(H(t,s)Q(s)-14h2(t,s)I)ds],
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
(3.6)A(t)≡0,B(t)=(t+1)2I2,C(t)=-[ρt200σ2t2],
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
(3.7)Ψ(s)=[ρt200σ2t2],P1(t,s,r)=-1(t-s)2(s-r)2(s+1)2I2.
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,
(3.8)lim⁡t→∞1t2α+1[∫rt(t-s)2(s-r)2α(ρt2-32(t-s)2(s-r)2(s+1)2)ds]=ρα(2α-1)(2α+1).
By Corollary 2.6, we find that system (3.1) is oscillatory for ρ>1/4.



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