# New Interval Oscillation Criteria for Certain Linear Hamiltonian Systems

Authors: Jing Shao; Fanwei Meng
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

Using a generalized Riccati transformation and the general integral means technique, some new interval oscillation criteria for the linear matrix Hamiltonian systemU'=(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.


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## Body

## 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:
(1)U′=(A(t)-λ(t)I)U+B(t)V,V′=C(t)U+(μ(t)I-A*(t))V,
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]≥⋯≥λn[M]. The trace of M is denoted by tr(M), and recall that tr(M)=∑k=1nλk(M).Whenλ(t)=μ(t)=0, the system (1) reduces to the following linear self-adjoint matrix Hamiltonian system:
(2)U′=A1(t)U+B1(t)V,V′=C1(t)U-A1*(t)V,t∈J.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.
Letℳ 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 functionH=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,
(3)∂H(t,s)∂t=h1(t,s)H(t,s),∂H(t,s)∂s=-h2(t,s)H(t,s),
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)=Hμ(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) functionq: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 existsf(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
(4)q{∫ab‍[z2D1-α(zA1*-z′I)B1-1(zA1-z′I)](t)dt}>0,
where D1(t)={α[-C1-2fK1+f2B1-1-(fB1-1)′]}(t) and K1(t)=(A1*B1-1+B1-1A1)(t). Then system (2) is oscillatory.Theorem 5.
Suppose that0<B1-1(t)≤I. If, for each r≥t0 and for some μ,ν>1,
(5)limsupt→∞1tμ+ν-1λ1λ×{∫rt(t-s)μ(s-r)νℊ11ℊ11g×(D¯0(s)+2νt-(μ+ν)s+μr(t-s)(s-r)K1(s))ds}>μν(μ+ν-2)Γ(μ-1)Γ(ν-1)4Γ(μ+ν),
where D¯0(s)=(-C1-A1*B1-1A1)(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 existsf(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
(6)q{∫ab[α(t)e-∫t0t‍(λ+μ)(s)ds(z2D)(t)mmmm-α(t)e-∫t0t‍(λ+μ)(s)ds[(z(A*-QB)-z′I)B-1mmmk-α(t)e-∫t0t‍(λ+μ)(s)dsa×(z(A-BQ)-z′I)B-1](t)α(t)e-∫t0t‍(λ+μ)(s)ds]dt∫ab}>0,
where
(7)D(t)={α[(e-∫‍(λ+μ)fB-1)′-e-∫‍(λ+μ)(QA-QBQ+C-(λ+μ)Q+A*Q)===l=-2fK+e-∫‍(λ+μ)f2B-1-(e-∫‍(λ+μ)fB-1)′]}(t),K(t)=e-∫t0t‍(λ+μ)(s)ds[(A*B-1+B-1A)(t)-2Q],
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
(8)W(t)=α(t)e-∫t0t‍(λ+μ)(s)ds=×[V(t)U-1(t)+f(t)B-1(t)+Q],t∈J.
Then W(t) is Hermitian and satisfies the Riccati equation
(9)W′(t)=-1α(t)e∫t0t‍(λ+μ)(s)ds=×[WBW-A*W-WA+QBW+WBQ-D](t).
Let E(t)=W(t)+α(t)e-∫t0t‍(λ+μ)(s)ds(B-1A-Q)(t). Then we get
(10)1αE*BE=1αW*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).
It follows from (9) and (10) that, for t∈J,
(11)W′=-1αe-∫‍(λ+μ)E*BE-D+αe-∫‍(λ+μ)A*B-1A-αe-∫‍(λ+μ)(AQ*+QA-QBQ).
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
(12)1αR-1G*GR-1=z2αE*(e∫‍(λ+μ)B)E-zz′(E*+E)=+αz′2(e∫‍(λ+μ)B)-1=z2αe∫‍(λ+μ)E*BE-zz′(E*+E)=+αe-∫‍(λ+μ)z′2B-1.
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
(13)∫abz2(t)D(t)dtα(t)≤∫abα(t)e-∫t0t‍(λ+μ)(s)dsα(t)αlk(t)×(z′2B-1z2(A*B-1A-A*Q-QA+QBQ)α(t)αlkkle-∫t0t‍--zz′(A*B-1+B-1A-2Q)+z′2B-1)(t)dtα(t)=∫ab‍α(t)e-∫t0t‍(λ+μ)(s)dsα(t)kle∫t0t‍×[(z(A-BQ)*-z′I)B-1(z(A-BQ)-z′I)](t)dt,
which implies that
(14)∫ab[z2D-αe-∫‍(λ+μ)(z(A*-QB)-z′I)z2D×B-1(z(A-BQ)-z′I)αe-∫‍(λ+μ)](t)dt≤0.
Applying the functional q to (14), we obtain a contradiction with assumption (6). This completes the proof.Corollary 7.
Suppose that there existsf(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
(15)λi{∫ab[z2D-αe-∫‍(λ+μ)(z(A*-QB)-z′I)====l=×B-1(z(A-BQ)-z′I)αe-∫‍(λ+μ)](t)dt∫ab[z2D-(αe-∫‍(λ+μ)(z(A*-QB)-z′I)}>0,
for some i∈{1,2,…,n}, where D(t) is defined as previously mentioned, then system (1) is oscillatory.Corollary 8.
Suppose that there existsf(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
(16)limsupt→∞q{∫rt‍(H2μ(t,s)H~2ν(s,r)D(s)-α(s)e-∫‍(λ+μ)=====kl===×Ψ*(t,s,r)B-1(s)Ψ(t,s,r)H~2ν)ds∫rt‍}>0,
where D(s) is defined as that in Theorem 6, and
(17)Ψ(t,s,r)=Hμ(t,s)H~ν(s,r)(A-BQ)(s)=-νHμ(t,s)H~ν-1/2(s,r)h~1(s,r)=+μh2(t,s)Hμ-1/2(t,s)H~ν(s,r),
then system (1) is oscillatory.LetH(t,s)=H~(t,s)=(t-s)2, μ=ν=1/2; we have the following useful corollary.Corollary 9.
Suppose that there existsf(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,
(18)limsupt→∞q{∫rt‍((t-s)2(s-r)2D(s)-α(s)e-∫‍(λ+μ)limsupt→∞q----×Ψ1*(t,s,r)B-1(s)Ψ1(t,s,r)α(s)e-∫‍(λ+μ))ds∫rt‍}>0,
where D(s) is defined as in Theorem 6, and
(19)Ψ1(t,s,r)=(t-s)(s-r)(A-BQ)(s)-(t+r-2s)I,
then system (1) is oscillatory.Theorem 10.
Suppose that0<B-1(t)≤e∫‍(λ+μ)I. If, for each r≥t0 and for some μ,ν>1,
(20)limsupt→∞1tμ+ν-1λ1{(D0(s)+νt-(μ+ν)s+μr2(t-s)(s-r)K(s))∫rt‍(t-s)μ(s-r)νlimsupt→∞m==lk×××(D0(s)+νt-(μ+ν)s+μr2(t-s)(s-r)K(s))ds}limsupt→∞>μν(μ+ν-2)Γ(μ-1)Γ(ν-1)4Γ(μ+ν),
where
(21)D0(s)=[-e-∫‍(λ+μ)(C+A*B-1A-(λ+μ)Q)](s),
and K(s) are defined as in Theorem 6, then system (1) is oscillatory.Proof.
We choosef(t)≡0, so a(t)≡1, and
(22)D(t)=[α(-e-∫‍(λ+μ)αααlα×(QA-QBQ+C-(λ+μ)Q+A*Q)-e-∫‍(λ+μ))α(-e-∫‍(λ+μ)](t).
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,
(23)∫rt‍z2(s)D0(s)dsz2(s)≤-∫rt‍(zz′K)(s)ds+∫rt‍z′2(s)e-∫‍(λ+μ)B-1(s)ds.
Suppose that z(s)=(t-s)μ/2(s-r)ν/2, we get
(24)u′(s)=z(s)νt-(μ+ν)s+μr2(t-s)(s-r).
So
(25)∫rt(t-s)μ(s-r)ν(D0(s)+νt-(μ+ν)s+μr2(t-s)(s-r)K(s))ds≤14≤14∫rt(t-s)μ-2(s-r)ν-2≤≤≤141414×[ν(t-s)-μ(s-r)]2e-∫‍(λ+μ)B-1(s)ds≤14≤14[∫rt‍ν2(t-s)μ(s-r)ν-2ds≤14≤14≤-2μν∫rt‍(t-s)μ-1(s-r)ν-1ds≤14≤14≤+μ2∫rt‍(t-s)μ-2(s-r)νds]I.
As in the proof of [11, Theorem 2.7], we obtain
(26)∫rt(t-s)μ(s-r)ν(D0(s)+νt-(μ+ν)s+μr2(t-s)(s-r)K(s))ds≤14≤14(t-r)μ+ν-1[μν2Γ(ν-1)Γ(μ)Γ(μ+ν)≤14(t---r)μμ+ν-1-2μνΓ(ν)Γ(μ)Γ(μ+ν)≤14(t---r)μμ+ν-1+μ2νΓ(ν)Γ(μ-1)Γ(μ+ν)]I14≤=(t-r)μμ+ν-1μν(μ+ν-2)Γ(μ-1)Γ(ν-1)4Γ(μ+ν)I.
Dividing both sides by tμ+ν-1 and taking the largest eigenvalue, we have
(27)limsupt→∞1tμ+ν-1λ1limsupsup×{∫rt‍(t-s)μ(s-r)νkmllimsupsupssupt→∞×(D0(s)+νt-(μ+ν)s+μr2(t-s)(s-r)K(s))ds}limsup≤μν(μ+ν-2)Γ(μ-1)Γ(ν-1)4Γ(μ+ν).
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
(28)A(t)=[01t-t-1t-t0],C(t)=[[γtcost+34t3+2t+t]00[γtcost+34t3-2t+t]],B(t)=tI2,t≥1,λ=1,μ=-1,Q=[0110].
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
(29)∫ab[z2D-αe-∫‍(λ+μ)(z(A-BQ)*-z′I)(zkm)×B-1(z(A-BQ)-z′I)](t)dt===∫-π/2+2kππ/2+2kπ[(-cost-sinttsintt-cost)γsin2tcostI2‍γsin2tcostcostI2-(-cost-sinttsintt-cost)γsin2tcostcostI2×(-costsintt-sintt-cost)]dt===[∫-π/2+2kππ/2+2kπ‍(γsin2tcost-cos2t-sin2tt2)dt]I2==≥[∫-π/2+2kππ/2+2kπ‍(γsin2tcost-cos2t-sin2t)dt]I2===[∫-π/2π/2‍sin2t(γcost-1)dt]I2==>0(forγ>12).
So we obtain that the Hamiltonian system is oscillatory for γ>1/2.



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