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

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Discrete Dynamics in Nature and Society Volume 2011, Article ID 659503,12pages doi:10.1155/2011/659503

Research Article

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

Yingxin Guo

^{1, 2}

and Junchang Wang

³

1College of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

2Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China

3Department of Mathematics, Shangqiu Normal University, Shangqiu, Henan 476000, China

Correspondence should be addressed to Junchang Wang,[email protected] Received 26 February 2011; Accepted 3 April 2011

Academic Editor: Mingshu Peng

Copyrightq2011 Y. Guo and J. Wang. 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.

By employing a generalized Riccati technique and an integral averaging technique, some new oscillation criteria are established for the second-order matrix diﬀerential systemU AxU BtV,VCxU−A^∗tV, whereAt,Bt, andCtaren×n-matrices, andB,Care 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

UAxUBtV,

VCxU−A^∗tV, t≥t0, 1.1

whereAt,Bt, andCtaren×n-matrices andB,Care Hermitian, that is,B^∗t Bt, C^∗t Ct. For any matrixA, the transpose ofAis denoted byA^∗.

For any real symmetric matrixesP,Q,R, we writeP≥Qmeaning thatP−Q≥0; that is,P−Qis positive semidefinite andP > Qmeaning thatP−Q >0; that is,P−Qis positive definite.

Definition 1.1. A solutionUt, Vtof1.1is called nontrivial if detUt/0 for at least one t≥t0.

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Definition 1.2. A nontrivial solution Ut, Vt of 1.1 is called prepared if U^∗tVt − V^∗tUT 0 for everyt≥t0.

Definition 1.3. System1.1 is called oscillatory ont0,∞if there is a nontrivial prepared solutionUt, Vtof1.1having the property that detUtvanishes onT,∞for every T > t0. Otherwise, it is called nonoscillatory.

Note 1. It follows from1, Theorem 8.1, page 303that if the system1.1 is oscillatory on t0,∞, then every nontrivial prepared solution Ut, Vt of 1.1has the property that detUtvanishes onT,∞for everyT > t0.

The oscillation problem for system1.1and its various particular cases such as the second-order matrix diﬀerential systems

YtQtYt 0, t∈t0,∞, 1.2 PtYtQtYt 0, t∈t0,∞, 1.3

has been studied extensively in recent years, for example, see 1–23. Some of the most important conditions that guarantee that system1.2is oscillatory are as follows:

lim_{t→ ∞}λ1{_t

t0Qsds}∞see4,6, lim_{t→ ∞}inf1/t_t

t0

_s

t0trQτdτ ds >−∞and lim_{t→ ∞}sup1/t_t

t0λ1_s

t0Qτdτds∞or

lim_{t→ ∞}sup1/t_t

t0{λ1_s

t0Qτdτ}²ds∞see5, lim_{t→ ∞}sup1/t^m−1λ1_t

t0t−s^m−1Qsdsds∞,m >2 is an integersee2.

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

Erbe, Kong, and Ruan’s Theorem

LetHt, sandht, sbe continuous onD{ft, s:t≥s≥t0}such thatHt, t 0 fort≥t0

andHt, s> 0 fort > s≥ t0. We assume further that the partial derivative∂/∂sHt, s Hst, sis nonpositive and continuous fort≥s≥t0andht, sis defined by

Hst, s −ht, sHt, s^1/2, t, s∈D. 1.4

Finally, we assume that

t→ ∞limsup 1 Ht, t0λ1

_t

t0

Ht, sQs−1

4h²t, sI

ds

∞, 1.5

where λ1A ≥ λ2A ≥ · · · ≥ λnA denotes the usual ordering of the eigenvalues of the symmetric matrixA;Iis then×nidentity matrix. Then system1.2is oscillatory.

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

Meng, Wang, and Zheng’s Theorem

LetHt, sandht, sbe continuous onD{t, s:t≥s≥t0}such thatHt, t 0 fort≥t0

andHt, s> 0 fort > s≥ t0. We assume further that the partial derivative∂/∂sHt, s Hst, sis nonpositive and continuous fort≥s≥t0andht, sis defined by

Hst, s −ht, sHt, s^1/2, t, s∈D. 1.6

If there exists a functionf∈C¹t0,∞such that

t→ ∞limsup 1 Ht, t0λ1

_t

t0

Ht, sRs−1

4ash²t, sI

ds

∞, 1.7

where at exp{−2

tfsds}, Rt at{Qt f²tI −ftI}. Then system1.2 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 t, s, land 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 2.1seem to be more complicated compared to the known ones, with appropriate choices of the functions , f, we derive a number of oscillation criteriasee also2.2, 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 Ht, s ∈ CD, R, whereD {t, s:t0 ≤ s≤t < ∞}, which satisfiesHt, t 0,Ht, s>0 fort > sand has partial derivative∂H/∂sonDsuch that

∂H

∂s −ht, sHt, s^1/2, 2.1

wherehis locally integrable with respect tosinD.

In this paper, let a function t, s, lbe continuous onD {t, s, l:t0≤l≤s≤t <

∞}, which satisfies t, t, l 0, t, s, l>0 forl≤s < tand has the partial derivative∂ /∂s onDsuch that∂ /∂sis locally integrable with respect tosinD, and we call the two positive numbersγandδadmissible22if they satisfy the conditionγδ >1.

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Theorem 2.1. If there exist a functionf∈C¹t0,∞and two admissible numbersγ,δsuch that

t→ ∞limsup 1

2t, t0, t0λ1

_t

t0

2t, s, t0Ψs γδPt, s, t0 ds

∞, 2.2

where Ψs bs−C−fA A^∗ f²B−fIs, I is the n×n identity matrix, bs exp−2_s

x0fςdς, and

Pt, s, t0 bs ²t, s, t0

fAA^∗−A^∗B⁻¹A s

−bs t, s, t0

st, s, t0−fs t, s, t0

×

A^∗B⁻¹B⁻¹A s

−bs

st, s, t0−fs t, s, t0

B^−1/2s−fs t, s, t0B^1/2s₂ ,

2.3

then system1.1is oscillatory.

Proof. Suppose to the contrary that system 1.1 is nonoscillatory. Then there exists a nontrivial prepared solutionUt, Vtof1.1such thatUtis nonsingular onT,∞for someT > t0. Without loss of generality, we may assume that detUt/0 fort≥t0. Define

Wt bt

VtU⁻¹t ftI

, t≥t0. 2.4

ThenWtis well defined, Hermitian, and it satisfies the Riccati equation

WWAA^∗W1

bWBW−fWBBW−2W Ψ

t 0 2.5

ont0,∞. Multiplying2.5, withtreplaced bys, by ²t, s, t0, integrating fromt0tot, and picking two admissible numbersγandδ, we obtain

_t

t0

2t, s, t0Ψsds− _t

t0

2t, s, t0Wsds− _t

t0

2t, s, t0

bs WBWsds

− _t

t0

2t, s, t0

WAA^∗W−fWBBW−2W sds

²t, t0, t0Wt0− _t

t0

2t, s, t0

bs WBWsds

− _t

t0

2t, s, t0

WAA^∗W−fWBBW sds

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2 _t

t0

t, s, t0

st, s, t0−fs t, s, t0

Wsds ²t, t0, t0Wt0−1

γ _t

t0

Q^∗Qt, s, t0ds−γδ _t

t0

Pt, s, t0ds

−γδ−1 γδ

_t

t0

2t, s, t0 bs

RW^∗RW sds,

2.6

whereRt

Btand

Qt, s, t0 t, s, t0

δbsRWs−γ

δbs t, s, t0

fR−R⁻¹A s

γ

δbs

st, s, t0−fs t, s, t0 R⁻¹s.

2.7

Then _t

t0

2t, s, t0Ψs γδPt, s, t0 ds ²t, t0, t0Wt0−1 γ

_t

t0

Q^∗Qt, s, t0ds

−γδ−1 γδ

_t

t0

2t, s, t0 bs

RW^∗RW sds

≤ ²t, t0, t0Wt0.

2.8

This implies that

λ1

_t

t0

2t, s, t0Ψs γδPt, s, t0 ds

≤ ²t, t0, t0λ1Wt0, 2.9

and then

1

2t, t0, t0λ1

t t0

2t, s, t0Ψs γδPt, s, t0 ds

≤λ1Wt0. 2.10

Taking the upper limit in both sides of 2.10 as t → ∞, the right-hand side is always bounded, which contradicts condition2.2. This completes the proof ofTheorem 2.1.

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

t→ ∞limsup 1

2t, t0, t0 _t

t0

2t, s, t0trΨs γδtrPt, s, t0 ds

∞, 2.11

whereΨs,bs, andPt, s, t0are as inTheorem 2.1, then system1.1is oscillatory.

By8, the trace tr : S → Ris a positive linear functional onS, where the spaceSis the linear space of all real symmetricn×nmatrices. And noting that two admissible numbers γ,δsatisfyingγδ >1, then we have the following corollary fromTheorem 2.2.

Corollary 2.3. If there exist a functionf∈C¹t0,∞and two admissible numbersγ,δsuch that

t→ ∞limsup 1

2t, t0, t0 _t

t0

2t, s, t0trΨs trPt, s, t0 ds

∞, 2.12

whereΨs,bs, andPt, s, t0are as inTheorem 2.1, then system1.1is oscillatory.

Proof. By virtue of a simple property of limits

t→ ∞limsup 1

2t, t0, t0 _t

t0

2t, s, t0trΨs γδtrPt, s, t0 ds

> lim

t→ ∞sup 1

2t, t0, t0 _t

t0

2t, s, t0trΨs trPt, s, t0 ds

2.13

and2.12, the conclusion follows fromTheorem 2.2.

If we choose t, s, t0

Ht, s/Ht, t0inTheorem 2.1, then

t, t0, t0

Ht, t0

Ht, t01, 2.14

we have the following.

Corollary 2.4. If there exist a functionf∈C¹t0,∞and two admissible numbersγ,δsuch that

tlim→ ∞sup 1 Ht, t0λ1

_t

t0

Ht, sΨ1s γδP1t, s ds

∞, 2.15

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whereHt, sare as in Erbe, Kong, and Ruan’s Theorem,Ψ1s bs−C−γδA^∗B⁻¹Af²B− fIγδfA^∗B⁻¹B⁻¹As,Iis then×nidentity matrix,bs exp−2_s

x0fςdς, and

P1t, s bsht, s Ht, s 2

−bs

ht, s

2 fs

Ht, s

B^−1/2s fs

Ht, sB^1/2s 2

,

2.16

Remark 2.5. In the last literature1–4,12,15,23, most oscillation results were given in the form of lim_{t→ ∞}sup1/Ht, t0λ1· ∞. Obviously,Theorem 2.1extends and improves a number of existing results and handles the cases not covered by known criteria, which can be seen fromCorollary 2.4.

If we chooseft 0 and let t, s, r

t−s^α/t−r^βforα, β >1/2 inTheorem 2.1, then we have the following.

Corollary 2.6. If there exist two real numbersα, β >1/2 and two admissible numbersγ,δsuch that

t→ ∞limsup 1 t^αλ1

_t

t0

t−s^α γδ

αt−s^α−1 2

A^∗B⁻¹B⁻¹A −A^∗B⁻¹A

−αt−s^2α−1

4 B⁻¹−γδC

ds

∞,

2.17

If we choose appropriatef inTheorem 2.1such thatbtB⁻¹t ≤ I fort ≥t0and let t, s, r

t−s^α/t−rforα >2, then we have the following

Corollary 2.7. If there exist a functionf∈C¹t0,∞and two admissible numbersγ,δsuch that for someα >1/2 and for everyr≥t0,

tlim→ ∞sup 1 t^2α1λ1

_t

r

t−s²s−r^2α

Ψs γδP1t, s, r ds

> α

2α−12α1, 2.18

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whereΨs,bsare as inTheorem 2.1and

P1t, s, r bs

fAA^∗−A^∗B⁻¹A s

− bs

t−ss−rrαt−α1s

bsfs

A^∗B⁻¹B⁻¹A−f

B2IB⁻¹ s

2bsfs

t−ss−rrαt−α1s IB⁻¹

− bs

t−s²s−r²B⁻¹,

2.19

Proof. Assume to the contrary that 1.1 is nonoscillatory. ThenUtis nonsingular for all suﬃciently larget, sayt≥T≥t0. Similar to the proof ofTheorem 2.1, fort≥T ≥t0, we have

_t

T

2t, s, TΨs γδP1t, s, T ds≤γδ _t

T

bsB⁻¹s

st, s, T2

ds

≤γδ _t

T

s−T^2α−1Tαt−α1s²I.

2.20

This implies that

λ1

_t

T

2t, s, TΨs γδP1t, s, T ds

≤ _t

T

s−T^2α−1Tαt−α1s²ds

α

2α−12α1t−T^2α1.

2.21

Then

t→ ∞limsup 1 t^2α1λ1

_t

T

2t, s, TΨs γδP1t, s, T ds

≤ α

2α−12α1, 2.22

which contradicts assumption2.18. This completes the proof ofCorollary 2.7.

WhenAt≡0,B⁻¹t Ptand−Ct Qtfort≥t0, then system1.1reduces to system1.3.

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

t→ ∞limsup 1

2t, t0, t0λ1

_t

t0

2t, s, t0Ψ1s γδP2t, s, t0 ds

∞, 2.23

where Ψ1s bsQt f²P⁻¹t − fIs, I is the n × n identity matrix, bs exp−2_s

x0fςdς, and P2t, s, t0 −bs

st, s, t0−fs t, s, t0

P^1/2s−fs t, s, t0P^−1/2s₂

, 2.24

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

Theorem 2.9. If there exist a functionf∈C¹t0,∞and two admissible numbersγ, δsuch that

tlim→ ∞supλ1

_t

t0

2t, s, t0trΨ1s γδtrP2t, s, t0 ds

∞, 2.25

whereΨ1s,bs, andP2t, s, t0are as inTheorem 2.8, then system1.3is oscillatory.

By8, the trace tr : S → Ris a positive linear functional onS, where the spaceSis the linear space of all real symmetricn×nmatrices. And noting that two admissible numbers γ,δsatisfyingγδ >1, then we have the following corollary fromTheorem 2.9.

Corollary 2.10. If there exist a functionf ∈C¹t0,∞and two admissible numbersγ,δsuch that

t→ ∞limsupλ1

_t

t0

2t, s, t0trΨ1s trP2t, s, t0 ds

∞, 2.26

whereΨ1s,bs, andP2t, s, t0are as inTheorem 2.8, then system1.3is oscillatory.

ByCorollary 2.7and1.3, we easily get the following theorem:

Theorem 2.11. If there exist a functionf ∈C¹t0,∞and two admissible numbersγ,δsuch that for someα >1/2 and for everyr≥t0,

t→ ∞limsup 1 t^2α1λ1

_t

r

t−s²s−r^2α

Ψ1s γδP3t, s, r ds

> α

2α−12α1, 2.27

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whereΨ1s,bsare as inTheorem 2.8and

P3t, s, r −bsf²s

P2IP⁻¹ s− bs

t−s²s−r²Ps

2bsfs

t−ss−rrαt−α1sIPs,

2.28

3. Examples

Example 3.1. Consider the Euler diﬀerential system

Ydiag n

t²,m t²

Y 0, t≥1, m≥n >0. 3.1

If we chooseft 0, thenat 1,Ψ1t diagn/t², m/t²andP3t, s, r 1/t−s²s− r²I. Note that for eachr≥1,

t→ ∞lim 1 t^2α1

_t

r

t−s²s−r^2α m

t² − γδ

t−s²s−r²

ds

lim

t→ ∞

1 t^2α1

_t

r

mt−s²s−r^2α

t² ds− lim

t→ ∞

γδ t^2α1

_t

r

s−r^2α−2ds

m

α2α−12α1.

3.2

Obviously, for anym >1/4, there existsα >1/2 such that m

α2α−12α1 > α

2α−12α1. 3.3 This means that2.25holds. By Theorem 2.11, we find that system3.1is oscillatory for m >1/4.

Remark 3.2. As pointed out in3, the above-mentioned criteria1.5of Erbe, Kong, and Ruan cannot be applied to the Euler diﬀerential system3.1, for

t→ ∞limsup 1 Ht,1λ1

_t

1

Ht, sQs−1

4h²t, sI

ds

≤ lim

t→ ∞

_t

1

m

s²dsm <∞. 3.4

Though the above-mentioned criteria 1.7of Meng, Wang, and Zheng’s Theorem can be applied to the Euler diﬀerential system, our results are sharper than theirs, which can be seen fromExample 3.1.

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Remark 3.3. It is interesting for the fact that If we choose ft 0, t, s, T Ht, s/Ht, T, then for diﬀerential system1.2, we have

t→ ∞limsupλ1

_t

1

2t, s,1Ψ1s P2t, s,1 ds

lim

t→ ∞sup 1

Ht,1λ1

_t

1

Ht, sQs−1

4h²t, sI

ds

,

3.5

whereHt, sare as in Erbe, Kong, and Ruan’s Theorem. Obviously,Theorem 2.8extends and improves a number of existing results and handles the cases not covered by known criteria.

Example 3.4. Consider the 4-dimensional system1.1where

At≡0, Bt t1²I2, Ct −

⎡

⎢⎣ ρ t² 0

0 σ

2t²

⎤

⎥⎦, 3.6

and whereρ≥ σ >0 andt≥1. If we letft 0, thenbt 1 andbtB⁻¹t≤ I2fort≥ 1.

Thus, we have

Ψs

⎡

⎢⎣ ρ t² 0

0 σ

2t²

⎤

⎥⎦, P1t, s, r − 1

t−s²s−r²s1²I2. 3.7

Thus, if we choose two admissible numbersγ,δsuch thatγδ 3/2, then for someα > 1/2 and for everyr ≥t0,

t→ ∞lim 1 t^2α1

_t

r

t−s²s−r^2α ρ

t² − 3

2t−s²s−r²s1²

ds

ρ

α2α−12α1. 3.8

ByCorollary 2.6, we find that system3.1is oscillatory forρ >1/4.

Acknowledgment

This wrok was supported financially by the NNSF of China10801088.

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