Lyapunov inequalities and stability for discrete linear Hamiltonian systems

Applied Mathematics and Computation 218 (2011) 574–582

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Lyapunov inequalities and stability for discrete linear Hamiltonian systems q Qi-ming Zhang a,b, X.H. Tang a,⇑ a b

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, PR China College of Science, Hunan University of Technology, Zhuzhou 412007, Hunan, PR China

a r t i c l e

i n f o

a b s t r a c t In this paper, we establish several new Lyapunov type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained. Ó 2011 Elsevier Inc. All rights reserved.

Keywords: Discrete Hamiltonian system Lyapunov inequality Generalized zero Stability

1. Introduction Consider a discrete linear Hamiltonian system

DxðnÞ ¼ aðnÞxðn þ 1Þ þ bðnÞyðnÞ;

DyðnÞ ¼ cðnÞxðn þ 1Þ  aðnÞyðnÞ;

ð1:1Þ

where a(n), b(n) and c(n) are real-valued functions defined on Z, D denotes the forward difference operator defined by Dx(n) = x(n + 1)  x(n). Throughout this paper, we always assume that

bðnÞ P 0;

8n 2 Z: >

ð1:2Þ

r

>

Let u(n) = (x(n), y(n)) , u (n) = (x(n + 1), y(n)) and

J¼



0

1

1 0

 ;

HðnÞ ¼





cðnÞ aðnÞ : aðnÞ bðnÞ

Then we can rewrite (1.1) as a standard discrete linear Hamiltonian system

DuðnÞ ¼ JHðnÞur ðnÞ:

ð1:3Þ

In recent papers [1–8], some dynamical behaviors of solutions of system (1.3) have been discussed, such as existence of periodic solutions, strong limit points, Weyl–Titchmarsh theory, spectral theory, eigenvalue problems and disconjugacy. For the second-order difference equation

D½pðnÞDxðnÞ þ qðnÞxðn þ 1Þ ¼ 0;

ð1:4Þ

where p(n) > 0, which has been investigated extensively and applied in literatures. If we let y(n) = p(n)Dx(n), then (1.4) can be written as an equivalent Hamiltonian system of type (1.1):

q This work is partially supported by the NNSF of China (No. 10771215) and by Scientific Research Fund of Hunan Provincial Education Department (No. 10C0655). ⇑ Corresponding author. E-mail address: [email protected] (X.H. Tang).

0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.101

Q.-m. Zhang, X.H. Tang / Applied Mathematics and Computation 218 (2011) 574–582

DxðnÞ ¼

1 yðnÞ; pðnÞ

DyðnÞ ¼ qðnÞxðn þ 1Þ;

575

ð1:5Þ

where

aðnÞ ¼ 0; bðnÞ ¼

1 ; pðnÞ

cðnÞ ¼ qðnÞ:

In the meantime, system (1.1) is also a discrete analogy of the following first-order polar Hamiltonian system

x0 ðtÞ ¼ aðtÞxðtÞ þ bðtÞyðtÞ;

y0 ðtÞ ¼ cðtÞxðtÞ  aðtÞyðtÞ:

ð1:6Þ

In the discrete case, instead of usual zero, we adopt the following concept of generalized zero, which is due to Hartman [9]. Definition 1.1 [9]. A function f : Z ! R is said to have a generalized zero at n0 2 Z provided either f(n0) = 0 or f(n0)f(n0 + 1) < 0. It is a classical topic for us to study Lyapunov type inequalities which have proved to be very useful in oscillation theory, disconjugacy, eigenvalue problems and numerous other applications in the theory of differential and difference equations. In recent years, there are many literatures which improved and extended the classical Lyapunov inequality for Hamiltonian systems including continuous and discrete cases. Here we only mention some references [9–22]. In the recent paper [18], Guseinov and Kaymakcalan researched system (1.1) and obtained some interesting Lyapunov type inequalities and stability criteria. Theorem 1.2 [18]. Let a; b 2 Z with a 6 b  2. Assume (1.1) has a real solution (x(n), y(n)) such that x(a) = x(b) = 0 and x(n) is not identically zero on [a, b]. Then one has the following inequality b2 X

" jaðnÞj þ

b1 X

n¼a

bðnÞ 

n¼a

b2 X

#1=2

cþ ðnÞ

P 2;

ð1:7Þ

n¼a

where and in the sequel

cþ ðnÞ ¼ maxfcðnÞ; 0g:

ð1:8Þ

Theorem 1.3 [18]. Suppose that

1  aðnÞ > 0;

bðnÞ > 0;

cðnÞ > 0; 8 n 2 Z

ð1:9Þ

and let a; b 2 Z with a 6 b  2. Assume (1.1) has a real solution (x(n), y(n)) such that x(n) has generalized zeros at a and b, and x(n) is not identically zero on [a, b], i.e.

xðaÞ ¼ 0 or xðaÞxða þ 1Þ < 0;

xðbÞ ¼ 0 or xðbÞxðb þ 1Þ < 0;

max jxðnÞj > 0:

a6n6b

ð1:10Þ

Then one has the following inequality b1 X

" jaðnÞj þ

n¼a

b X

bðnÞ 

n¼a

b1 X

#1=2

cðnÞ

> 1:

ð1:11Þ

n¼a

When a(n), b(n) and c(n) are N-periodic functions, by applying Floquet theory and the above theorems, Guseinov and Kaymakcalan [18] also obtained the following stability criterion. Theorem 1.4 [18]. Assume that

aðn þ NÞ ¼ aðnÞ; bðn þ NÞ ¼ bðnÞ; cðn þ NÞ ¼ cðnÞ; 8 n 2 Z; bðnÞ > 0; cðnÞ > 0; bðnÞcðnÞ  a2 ðnÞ P 0ð – 0Þ; 8 n 2 Z

ð1:12Þ ð1:13Þ

and N X n¼1

(" jaðnÞj þ

b0 þ

N X n¼1

# bðnÞ

N X

)1=2

cðnÞ

6 1;

ð1:14Þ

n¼1

where b0 = max16n6Nb(n). Then system (1.1) is stable. In the case a and b are usual zeros of x(n), i.e. x(a) = x(b) = 0, we remark that the bound 2 in the right side of (1.7) of Theorem 1.2 is the best possible constant in some sense, see [13]. However, when a or b is the generalized zero of x(n), the corresponding bound reduces to constant 1, see (1.11) in Theorem 1.3. In this paper, by using some simpler methods different from [18], we obtain a better Lyapunov type inequality than (1.11)

576

Q.-m. Zhang, X.H. Tang / Applied Mathematics and Computation 218 (2011) 574–582 b1 X

jaðnÞj þ

" b X

n¼a

b1 X

bðnÞ

n¼a

#1=2 þ

c ðnÞ

P 2;

ð1:15Þ

n¼a

only under the assumption

1  aðnÞ > 0;

8n 2 Z:

ð1:16Þ

Furthermore, applying the theorems obtained in this paper, the stability conditions (1.13) and (1.14) are greatly improved by the following conditions: there exists a non-negative function h(n) such that

jaðnÞj 6 hðnÞbðnÞ; n 2 ½1; N; N X



ð1:17Þ



cðnÞ  h2 ðnÞbðnÞ > 0

ð1:18Þ

n¼1

and N X

jaðnÞj þ

" N X

n¼1

bðnÞ

n¼1

N X

#1=2 þ

c ðnÞ

< 2:

ð1:19Þ

n¼1

2. Lyapunov type inequalities In this section, we establish some new Lyapunov type inequalities. Theorem 2.1. Suppose that (1.16) holds and let a; b 2 Z with a 6 b  1. Assume (1.1) has a real solution (x(n), y(n)) such that (1.10) holds. Then one has the following inequality b1 X

jaðnÞj þ

" b X

n¼a

bðnÞ

n¼a

b1 X

#1=2

cþ ðnÞ

P 2:

ð2:1Þ

n¼a

Proof. It follows from (1.10) that there exist n, g 2 [0, 1) such that

ð1  nÞxðaÞ þ nxða þ 1Þ ¼ 0

ð2:2Þ

ð1  gÞxðbÞ þ gxðb þ 1Þ ¼ 0:

ð2:3Þ

and

Multiplying the first equation of (1.1) by y(n) and the second one by x(n + 1), and then adding, we get

D½xðnÞyðnÞ ¼ bðnÞy2 ðnÞ  cðnÞx2 ðn þ 1Þ:

ð2:4Þ

Summing the Eq. (2.4) from a to b  1, we can obtain

xðbÞyðbÞ  xðaÞyðaÞ ¼

b1 X

bðnÞy2 ðnÞ 

n¼a

b1 X

cðnÞx2 ðn þ 1Þ:

ð2:5Þ

n¼a

From the first equation of (1.1), we have

½1  aðnÞxðn þ 1Þ ¼ xðnÞ þ bðnÞyðnÞ:

ð2:6Þ

Combining (2.6) with (2.2), we have

xðaÞ ¼ 

nbðaÞ yðaÞ: 1  ð1  nÞaðaÞ

ð2:7Þ

Similarly, it follows from (2.6) and (2.3) that

xðbÞ ¼ 

gbðbÞ yðbÞ: 1  ð1  gÞaðbÞ

ð2:8Þ

Substituting (2.7) and (2.8) into (2.5), we have b1 X

bðnÞy2 ðnÞ 

n¼a

which implies that

b1 X n¼a

cðnÞx2 ðn þ 1Þ ¼ 

gbðbÞ nbðaÞ y2 ðbÞ þ y2 ðaÞ; 1  ð1  gÞaðbÞ 1  ð1  nÞaðaÞ

577

Q.-m. Zhang, X.H. Tang / Applied Mathematics and Computation 218 (2011) 574–582 b1 b1 X X ð1  nÞ½1  aðaÞ gbðbÞ bðnÞy2 ðnÞ þ cðnÞx2 ðn þ 1Þ: bðaÞy2 ðaÞ þ y2 ðbÞ ¼ 1  ð1  nÞaðaÞ 1  ð1  gÞaðbÞ n¼a n¼aþ1

ð2:9Þ

Denote that

ð1  nÞ½1  aðaÞ ~ bðaÞ ¼ bðaÞ; 1  ð1  nÞaðaÞ

~ bðbÞ ¼

g 1  ð1  gÞaðbÞ

ð2:10Þ

bðbÞ

and

~ bðnÞ ¼ bðnÞ;

a þ 1 6 n 6 b  1:

ð2:11Þ

Then we can rewrite (2.9) as b X

2 ~ bðnÞy ðnÞ ¼

b1 X

n¼a

cðnÞx2 ðn þ 1Þ:

ð2:12Þ

n¼a

On the other hand, summing the first equation of (1.1) from a to s  1 and using (2.7), we obtain s1 X

xðsÞ ¼ xðaÞ þ

aðnÞxðn þ 1Þ þ

s1 X

n¼a

¼

s1 X

bðnÞyðnÞ ¼ 

n¼a

aðnÞxðn þ 1Þ þ

n¼a

s1 X

~ bðnÞyðnÞ;

s1 s1 X X nbðaÞ aðnÞxðn þ 1Þ þ bðnÞyðnÞ yðaÞ þ 1  ð1  nÞaðaÞ n¼a n¼a

a þ 1 6 s 6 b:

ð2:13Þ

n¼a

Similarly, summing the first equation of (1.1) from s to b  1 and using (2.8), we have b1 X

xðsÞ ¼ xðbÞ 

aðnÞxðn þ 1Þ 

b1 X

n¼s

¼

b1 X

bðnÞyðnÞ ¼ 

n¼s

aðnÞxðn þ 1Þ 

n¼s

b X

~ bðnÞyðnÞ;

b1 b1 X X gbðbÞ yðbÞ  aðnÞxðn þ 1Þ  bðnÞyðnÞ 1  ð1  gÞaðbÞ n¼s n¼s

a þ 1 6 s 6 b:

ð2:14Þ

n¼s

It follows from (2.13) and (2.14) that

jxðsÞj 6

s1 X

jaðnÞjjxðn þ 1Þj þ

n¼a

s1 X

~ bðnÞjyðnÞj;

aþ16s6b

~ bðnÞjyðnÞj;

a þ 1 6 s 6 b:

n¼a

and

jxðsÞj 6

b1 X

jaðnÞjjxðn þ 1Þj þ

n¼s

b X n¼s

Adding the above two inequalities, we have

2jxðsÞj 6

b1 X

b X

jaðnÞjjxðn þ 1Þj þ

n¼a

~ bðnÞjyðnÞj;

a þ 1 6 s 6 b:

ð2:15Þ

n¼a

Let jx(s⁄)j = maxa+16n6bjx(n)j. Applying the Cauchy inequality and using (2.12), we have 

2jxðs Þj 6

b1 X

jaðnÞjjxðn þ 1Þj þ

n¼a

b X

~ bðnÞjyðnÞj 6 jxðs Þj

n¼a



¼ jxðs Þj

b1 X

jaðnÞj þ

n¼a

" b X

jaðnÞj þ

" b X

n¼a

~ bðnÞ

n¼a

b1 X

b1 X

n¼a

#1=2 2

cðnÞx ðn þ 1Þ

n¼a

~ bðnÞ



6 jxðs Þj

8 b1
b X

#1=2 2 ~ bðnÞy ðnÞ

n¼a

" jaðnÞj þ

b X n¼a

~ bðnÞ

#1=2 9 = c ðnÞ : ; n¼a

b1 X

þ

ð2:16Þ

Dividing the latter inequality of (2.16) by jx(s⁄)j, we obtain b1 X

" jaðnÞj þ

n¼a

b X n¼a

~ bðnÞ

b1 X

#1=2 þ

c ðnÞ

P 2:

n¼a

Since

~ bðnÞ 6 bðnÞ;

a 6 n 6 b;

then it follows from (2.17) that (2.1). h In the case x(b) = 0, i.e. g = 0, we have the following equation

ð2:17Þ

578

Q.-m. Zhang, X.H. Tang / Applied Mathematics and Computation 218 (2011) 574–582 b1 X

2 ~ bðnÞy ðnÞ ¼

b2 X

n¼a

cðnÞx2 ðn þ 1Þ

ð2:18Þ

n¼a

and inequality

2jxðsÞj 6

b2 X

jaðnÞjjxðn þ 1Þj þ

n¼a

b1 X

~ bðnÞjyðnÞj;

aþ16s6b1

ð2:19Þ

n¼a

instead of (2.12) and (2.15), respectively. Similar to the proof of (2.17), we have b2 X

jaðnÞj þ

" b1 X

n¼a

~ bðnÞ

n¼a

b2 X

#1=2 þ

c ðnÞ

P 2:

ð2:20Þ

n¼a

~ Since bðnÞ 6 bðnÞ for a 6 n 6 b, it follows that b2 X

jaðnÞj þ

" b1 X

n¼a

bðnÞ

n¼a

b2 X

#1=2

cþ ðnÞ

P 2:

ð2:21Þ

n¼a

Therefore, we have the following theorem. Theorem 2.2. Suppose that (1.16) holds and let a; b 2 Z with a 6 b  2. Assume (1.1) has a real solution (x(n), y(n)) such that x(a) = 0 or x(a)x(a + 1) < 0 and x(b) = 0 and x(n) is not identically zero on [a, b]. Then inequality (2.21) holds. h

Remark 2.3. We obtain the same Lyapunov type inequality (2.21) as (1.7) under weaker assumptions than the ones of Theorem 1.2, which improves greatly the following inequality b2 X

jaðnÞj þ

" b1 X

n¼a

bðnÞ

n¼a

b2 X

#1=2

cþ ðnÞ

>1

n¼a

obtained in [18, Theorem 1.4]. Theorem 2.4. Suppose that (1.16) holds and let a; b 2 Z with a 6 b  1. Assume (1.1) has a real solution (x(n), y(n)) such that x(a) = 0 or x(a)x(a + 1) < 0 and (x(b), y(b)) = (kx(a), ly(a)) with 0 < kl 6 l2 6 1 and x(n) is not identically zero on [a, b]. Then one has the following inequality b1 X

jaðnÞj þ

" b1 X

n¼a

bðnÞ

n¼a

b1 X

#1=2 þ

c ðnÞ

P 2:

ð2:22Þ

n¼a

Proof. It follows from the assumption x(a) = 0 or x(a)x(a + 1) < 0 that there exists n 2 [0, 1) such that (2.2) holds. Further, by the proof of Theorem 2.1, (2.4)–(2.7) hold. Since (x(b), y(b)) = (kx(a), ly(a)), then by (2.5), we have

ðkl  1ÞxðaÞyðaÞ ¼

b1 X

bðnÞy2 ðnÞ 

n¼a

b1 X

cðnÞx2 ðn þ 1Þ:

ð2:23Þ

n¼a

Substituting (2.7) into (2.23), we have b1 X

bðnÞy2 ðnÞ 

n¼a

b1 X

cðnÞx2 ðn þ 1Þ ¼

n¼a

ð1  klÞnbðaÞ 2 y ðaÞ; 1  ð1  nÞaðaÞ

which implies that

j1 bðaÞy2 ðaÞ þ

b1 X n¼aþ1

bðnÞy2 ðnÞ ¼

b1 X

cðnÞx2 ðn þ 1Þ;

ð2:24Þ

n¼a

where

j1 ¼

1  ð1  klÞn  ð1  nÞaðaÞ : 1  ð1  nÞaðaÞ

On the other hand, summing the first equation of (1.1) from a to s  1 and using (2.7), we obtain

ð2:25Þ

Q.-m. Zhang, X.H. Tang / Applied Mathematics and Computation 218 (2011) 574–582

xðsÞ ¼ xðaÞ þ

s1 X

aðnÞxðn þ 1Þ þ

n¼a

s1 X

bðnÞyðnÞ ¼ 

n¼a

579

s1 s1 X X n aðnÞxðn þ 1Þ þ bðnÞyðnÞ bðaÞyðaÞ þ 1  ð1  nÞaðaÞ n¼a n¼a

s1 s1 X X ð1  nÞ½1  aðaÞ aðnÞxðn þ 1Þ þ bðnÞyðnÞ; bðaÞyðaÞ þ ¼ 1  ð1  nÞaðaÞ n¼a n¼aþ1

a þ 1 6 s 6 b:

ð2:26Þ

Similarly, summing the first equation of (1.1) from s to b  1 and using (2.7) and the fact that x(b) = kx(a), we have

xðsÞ ¼ xðbÞ 

b1 X

aðnÞxðn þ 1Þ 

n¼s

¼

b1 X

bðnÞyðnÞ ¼ kxðaÞ 

n¼s

b1 X

aðnÞxðn þ 1Þ 

n¼s

b1 X

bðnÞyðnÞ

n¼s

b1 b1 X X kn aðnÞxðn þ 1Þ  bðnÞyðnÞ; bðaÞyðaÞ  1  ð1  nÞaðaÞ n¼s n¼s

a þ 1 6 s 6 b:

ð2:27Þ

It follows from (2.26) and (2.27) that

jxðsÞj 6

s1 s1 X X ð1  nÞ½1  aðaÞ jaðnÞjjxðn þ 1Þj þ bðnÞjyðnÞj; bðaÞjyðaÞj þ 1  ð1  nÞaðaÞ n¼a n¼aþ1

jxðsÞj 6

b1 b1 X X jkjn jaðnÞjjxðn þ 1Þj þ bðnÞjyðnÞj; bðaÞjyðaÞj þ 1  ð1  nÞaðaÞ n¼s n¼s

aþ16s6b

and

a þ 1 6 s 6 b:

Adding the above two inequalities, we have

2jxðsÞj 6 j2 bðaÞjyðaÞj þ

b1 X

b1 X

jaðnÞjjxðn þ 1Þj þ

n¼a

a þ 1 6 s 6 b;

bðnÞjyðnÞj;

ð2:28Þ

n¼aþ1

where

j2 ¼

1  ð1  jkjÞn  ð1  nÞaðaÞ : 1  ð1  nÞaðaÞ

ð2:29Þ

Let jx(s⁄)j = maxa+16n6bjx(n)j. Applying (2.24), (2.28) and the Cauchy inequality, we have

2jxðs Þj 6 j2 bðaÞjyðaÞj þ

b1 X

jaðnÞjjxðn þ 1Þj þ

n¼a

6 jxðs Þj

b1 X



¼ jxðs Þj

" jaðnÞj þ " jaðnÞj þ

n¼a



6 jxðs Þj

b1 X

" jaðnÞj þ

n¼a

8 " b1
bðnÞjyðnÞj

n¼aþ1

n¼a b1 X

b1 X

!

b1 X j22 bðaÞ þ bðnÞ j1 n¼aþ1 b1 X

j bðaÞ þ bðnÞ j1 n¼aþ1 2 2

j1 bðaÞy2 ðaÞ þ

b1 X

!#1=2 bðnÞy2 ðnÞ

n¼aþ1

!

b1 X

#1=2 2

cðnÞx ðn þ 1Þ

n¼a

!

b1 b1 X X j22 bðaÞ þ bðnÞ cþ ðnÞx2 ðn þ 1Þ j1 n¼a n¼aþ1

!

b1 b1 X X j22 bðaÞ þ bðnÞ cþ ðnÞ j1 n¼a n¼aþ1

#1=2

#1=2 9 = : ;

ð2:30Þ

Dividing the latter inequality of (2.30) by jx(s⁄)j, we obtain b1 X

" jaðnÞj þ

n¼a

!

#1=2

b1 b1 X X j22 bðaÞ þ bðnÞ cþ ðnÞ j1 n¼a n¼aþ1

P 2:

Set d = 1  (1  n)a(a). Then d > 0. Since (1  n)[1  a(a)] P 0, it follows that n 6 d, and so

½d  ð1  jkjÞn2 6 d½d  ð1  klÞn: This, together with (2.25) and (2.29), implies that

j22 ¼ j1

h

1ð1jkjÞnð1nÞaðaÞ 1ð1nÞaðaÞ

i2

1ð1klÞnð1nÞaðaÞ 1ð1nÞaðaÞ

2

¼

½d  ð1  jkjÞn 6 1: d½d  ð1  klÞn

ð2:31Þ

580

Q.-m. Zhang, X.H. Tang / Applied Mathematics and Computation 218 (2011) 574–582

Substituting this into (2.31), we obtain (2.22). h

3. Stability criteria In this section, we discuss the stability for solutions of system (1.1) by applying the Lyapunov type inequalities obtained in the last section. To this end, we assume that system (1.1) is N-periodic, i.e. the coefficients a(n), b(n) and c(n) satisfy the periodicity conditions

aðn þ NÞ ¼ aðnÞ; bðn þ NÞ ¼ bðnÞ; cðn þ NÞ ¼ cðnÞ; 8n 2 Z:

ð3:1Þ

Definition 3.1. System (1.1) is said to be stable if all solutions are bounded on Z, unstable if all non-zero solutions are unbounded on Z, and conditionally stable if there exists a non-zero solution bounded on Z. Let u(n) = (x(n), y(n))> and

0 AðnÞ ¼ @

1

bðnÞ 1aðnÞ

1 1aðnÞ

bðnÞcðnÞ  1cðnÞ aðnÞ 1  aðnÞ  1aðnÞ

A:

ð3:2Þ

Then detA(n) = 1 for n 2 Z and system (1.1) can be written as

uðn þ 1Þ ¼ AðnÞuðnÞ:

ð3:3Þ

Let U(n) with U(0) = I2 be a fundamental matrix solution of (3.3). Then

Uðn þ 1Þ ¼ AðnÞUðnÞ;

Uðn þ NÞ ¼ UðnÞUðNÞ;

8n 2 Z:

ð3:4Þ

It follows that detU(n) = detU(0) = 1 for n 2 Z. The Floquet multipliers (real or complex) of (3.3) are the roots of

detðkI2  UðNÞÞ ¼ 0; which is equivalent to

k2  qk þ 1 ¼ 0;

ð3:5Þ

where q is the trace of matrix U(N). Let k1 and k2 are the Floquet multipliers, then we have

k1 þ k2 ¼ q;

k1 k2 ¼ 1:

It follows from the Floquet theory [23] that corresponding to each (complex) root kk there is a solution uk(n) = (xk(n), yk(n))> of (3.3) (or (1.1)) with xk(n) – 0 such that

uk ðn þ NÞ ¼ kk uk ðnÞ;

k ¼ 1; 2;

8n 2 Z:

ð3:6Þ

These are the so-called Floquet solutions of (3.3) (or (1.1)). Lemma 3.2 [18]. System (1.1) (or (3.3)) is unstable if jqj > 2 and stable if jqj < 2. Lemma 3.3. Assume that (3.1) holds, and that there exists a non-negative function h(n) such that (1.17) and (1.18) hold. If

q2 P 4, then system (1.1) has a non-zero solution (x(n), y(n)) such that x(n) has a generalized zero in [1, N]. Proof. Suppose that jqj P 2. Then one has real Floquet multipliers kk and real Floquet solutions uk(n) = (xk(n), yk(n))>, k = 1, 2. Let us consider any Floquet solution, say u1(n) = (x1(n), y1(n))>. We assert that x1(n) must have at least one generalized zero in the segment [1, N]. Otherwise, one may assume that x1(n) > 0 for n 2 [1, N] and so x1(n) > 0 for n 2 Z. Define z(n) :¼ y1(n)/x1(n). Due to (3.6), one sees that z(n) is N-periodic. From (1.1), we have

x1 ðnÞDy1 ðnÞ  y1 ðnÞDx1 ðnÞ cðnÞx1 ðnÞx1 ðn þ 1Þ  aðnÞ½x1 ðnÞ þ x1 ðn þ 1Þy1 ðnÞ  bðnÞy21 ðnÞ ¼ x1 ðnÞx1 ðn þ 1Þ x1 ðnÞx1 ðn þ 1Þ     y ðnÞ y1 ðnÞ y ðnÞ y1 ðnÞ ¼ cðnÞ  aðnÞ 1 þ  bðnÞ 1 x1 ðnÞ x1 ðn þ 1Þ x1 ðnÞ x1 ðn þ 1Þ     y1 ðnÞ y1 ðnÞ ¼ cðnÞ  aðnÞ zðnÞ þ  bðnÞzðnÞ : x1 ðn þ 1Þ x1 ðn þ 1Þ

DzðnÞ ¼

Since x1(n) > 0 for all n 2 Z, then it follows from (2.6) that

ð3:7Þ

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1 þ bðnÞzðnÞ ¼ 1 þ bðnÞ

y1 ðnÞ x1 ðn þ 1Þ ¼ ½1  aðnÞ > 0; x1 ðnÞ x1 ðnÞ

ð3:8Þ

which yields

y1 ðnÞ ½1  aðnÞzðnÞ : ¼ x1 ðn þ 1Þ 1 þ bðnÞzðnÞ

ð3:9Þ

Substituting (3.9) into (3.7), we obtain

DzðnÞ ¼ cðnÞ þ

  2aðnÞ þ a2 ðnÞ zðnÞ  bðnÞz2 ðnÞ : 1 þ bðnÞzðnÞ

ð3:10Þ

If b(n) > 0, then it is easy to verify that

  2aðnÞ þ a2 ðnÞ zðnÞ  bðnÞz2 ðnÞ a2 ðnÞ 6 h2 ðnÞbðnÞ: 6 bðnÞ 1 þ bðnÞzðnÞ

ð3:11Þ

If b(n) = 0, then it follows from (1.17) that a(n) = 0, hence

  2aðnÞ þ a2 ðnÞ zðnÞ  bðnÞz2 ðnÞ ¼ 0 ¼ h2 ðnÞbðnÞ: 1 þ bðnÞzðnÞ

ð3:12Þ

Combining (3.11) with (3.12), we have

  2aðnÞ þ a2 ðnÞ zðnÞ  bðnÞz2 ðnÞ 6 h2 ðnÞbðnÞ: 1 þ bðnÞzðnÞ

ð3:13Þ

Substituting (3.13) into (3.10), we obtain

DzðnÞ 6 cðnÞ þ h2 ðnÞbðnÞ:

ð3:14Þ

Summing it from 1 to N and noticing that z(n) is N-periodic, we obtain

06

N X 



cðnÞ  h2 ðnÞbðnÞ

n¼1

a contradiction with condition (1.18). h Theorem 3.4. Assume that 3.1, 1.17, 1.18 and 1.19 hold. Then system (1.1) is stable. Proof. Since (1.17) and (1.18), if jqj P 2, then one has real Floquet multipliers kk and real Floquet solutions uk(n) = (xk(n), yk(n))> such that (3.6) holds for k = 1, 2. Since k1k2 = 1, then 0 < minfk21 ; k22 g 6 1. Suppose k21 6 1. Then by Lemma 3.3, system (1.1) has a non-zero solution (x1(n), y1(n)) such that x1(n) has a generalized zero in [1, N], say n1. It follows from (3.6) that n1 + N is also a generalized zero of x1(n) and

ðx1 ðn1 þ NÞ; y1 ðn1 þ NÞÞ ¼ k1 ðx1 ðn1 Þ; y1 ðn1 ÞÞ: Applying Theorem 2.4 to the solution (x1(n), y1(n)) with a = n1, b = n1 + N and k = l = k1, we get n1X þN1

" jaðnÞj þ

n1X þN1

n¼n1

bðnÞ

n1X þN1

n¼n1

#1=2

cþ ðnÞ

P 2:

ð3:15Þ

n¼n1

Next, note that for any periodic function f(n) on Z with period N, the equation n0X þN1

f ðnÞ ¼

N X

n¼n0

f ðnÞ

n¼1

holds for all n0 2 Z. It follows from (3.15) that N X n¼1

" jaðnÞj þ

N X n¼1

bðnÞ

N X

#1=2 þ

c ðnÞ

P 2;

ð3:16Þ

n¼1

which contradicts condition (1.19). Thus jqj < 2 and hence system (1.1) is stable. h Applying Theorem 3.4 to the second-order difference Eq. (1.4), we can obtain the following classical stability result.

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Q.-m. Zhang, X.H. Tang / Applied Mathematics and Computation 218 (2011) 574–582

Corollary 3.5. Assume that

pðn þ NÞ ¼ pðnÞ > 0;

qðn þ NÞ ¼ qðnÞ;

8n 2 Z

ð3:17Þ

and

0<

N X

qðnÞ 6

n¼1

N X n¼1

qþ ðnÞ <

4 N P n¼1

:

ð3:18Þ

1 pðnÞ

Then Eq. (1.4) is stable. Acknowledgments The authors thank the referees for valuable comments and suggestions. References [1] S. Aubry, Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems, Review Article Physica D: Nonlinear Phenomena 216 (2006) 1–30. [2] M. Bohner, Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl 199 (1996) 804–826. [3] M. Bohner, Discrete linear Hamiltonian eigenvalue problems, Comput. Math. Appl. 36 (1998) 179–192. [4] L.H. Erbe, P.X. Yan, Disconjugacy for linear Hamiltonian difference systems, J. Math. Anal. Appl. 167 (1992) 355–367. [5] Y.M. Shi, Spectral theory of discrete linear Hamiltonian systems, J. Math. Anal. Appl 289 (2004) 554–570. [6] Y.M. Shi, Weyl–Titchmarsh theory for a class of discrete linear Hamiltonian systems, Linear Algebra Appl. 416 (2006) 452–519. [7] H.Q. Sun, Y.M. Shi, Strong limit point criteria for a class of singular discrete linear Hamiltonian systems, J. Math. Anal. Appl. 336 (2007) 224–242. [8] J.S. Yu, H.H. Bin, Z.M. Guo, Multiple periodic solutions for discrete Hamiltonian systems, Nonlinear Anal 66 (2007) 1498–1512. [9] P. Hartman, Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity, Trans. Amer. Math. Soc. 246 (1978) 1– 30. [10] R. Agarwal, C.D. Ahlbrandt, M. Bohner, A.C. Peterson, Discrete linear Hamiltonian systems: a survey, Dynam. Syst. Appl. 8 (1999) 307–333. [11] C.D. Ahlbrandt, A.C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Academic, Boston, MA, 1996. [12] M. Bohner, S. Clark, J. Ridenhour, Lyapunov inequalities on time scales, J. Inequal. Appl. 7 (2002) 61–77. [13] S.S. Cheng, A discrete analogue of the inequality of Lyapunov, Hokkaido Math. J. 12 (1983) 105–112. [14] S.S. Cheng, Lyapunov inequalities for differential and difference equations, Fasc. Math. 23 (1991) 25–41. [15] S. Clark, D.B. Hinton, Discrete Lyapunov inequalities for linear Hamiltonian systems, Math. Inequal. Appl. 1 (1998) 201–209. [16] S. Clark, D.B. Hinton, Discrete Lyapunov inequalities, Dynam. Syst. Appl. 8 (1999) 369–380. [17] G. Sh. Guseinov, A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems, J. Math. Anal. Appl. 335 (2007) 1195–1206. [18] G. Sh. Guseinov, B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. Math. Appl. 45 (2003) 1399–1416. [19] L.Q. Jiang, Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl. 310 (2005) 579–593. [20] M.G. Krein, Foundations of the theory of k-zones of stability of canonical system of linear differential equations with periodic coefficients. In Memory of A.A. Andronov, Izdat. Acad. Nauk SSSR, Moscow, 1955, pp. 413–498 (Amer. Math. Soc. Transl. Ser. 2, 120 (1983) 1–70). [21] X. Wang, Stability criteria for linear periodic Hamiltonian systems, J. Math. Anal. Appl 367 (2010) 329–336. [22] V.A. Yakubovich, V.M. Starzhinsky, Linear Differential Equations with Periodic Coefficients, Parts I and II, Wiley, New York, 1975. [23] S.N. Elaydi, An Introduction to Difference Equations, third ed., Springer-Verlag, New York, 2004.