Lyapunov-Type Inequalities for the Quasilinear Difference Systems

(1)

Discrete Dynamics in Nature and Society Volume 2012, Article ID 860598,16pages doi:10.1155/2012/860598

Research Article

Lyapunov-Type Inequalities for the Quasilinear Difference Systems

Qi-Ming Zhang

¹

and X. H. Tang

²

1College of Science, Hunan University of Technology, Zhuzhou, Hunan 412000, China

2School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Correspondence should be addressed to Qi-Ming Zhang,[email protected] Received 22 October 2011; Accepted 4 December 2011

Academic Editor: Zengji Du

Copyrightq2012 Q.-M. Zhang and X. H. Tang. 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.

We establish several Lyapunov-type inequalities for quasilinear diﬀerence systems, which generalize or improve all related existing ones. Applying these results, we also obtain some lower bounds for the first eigencurve in the generalized spectra.

1. Introduction

In 1964, Atkinson1investigated the following boundary value problem:

ΔrnΔun λqnun 1 1.1

with Dirichlet boundary condition:

ua ub 0, un/≡0, ∀n∈Za, b, 1.2

and he proved that boundary value problem1.1with1.2has exactlyb−a−1 real and simple eigenvalues, which can be arranged in the increasing order

λ1< λ2<· · ·< λ_b−a−1, 1.3

wherea, b ∈ Zwitha ≤ b−2,λ ∈ R,rn > 0 andqn > 0 for alln ∈ Z. Here and in the sequel,Za, b {a, a 1, a 2, . . . , b−1, b}.

(2)

In 1983, Cheng2proved that if the second-order diﬀerence equation

Δ²un qnun 1 0 1.4

has a real solutionunsuch that

ua ub 0, un/≡0, n∈Za, b, 1.5

then one has the following inequality

Fb−a^b−2

naqn≥4, 1.6

whereqn≥0 for alln∈Z, and

Fm

⎧⎪

⎨

⎪⎩ m²−1

m , if m−1 is even, m, if m−1 is odd,

1.7

and the constant 4 in1.6cannot be replaced by a larger number. Inequality1.6is a discrete analogy of the following so-called Lyapunov inequality:

b−a _b

a

qtdt >4, 1.8

if Hill’s equation

ut qtut 0 1.9

has a real solutionutsuch that

ua ub 0, ut/0, ∀t∈a, b, 1.10

whereqtis a real-valued continuous function defined onR,a, b ∈Rwitha < b. Equation 1.8was first established by Liapounoﬀ 3in 1907.

In 2008, ¨Unal et al.4established the following Lyapunov-type inequality:

b−1 na

1

rn^1/p−1

1−1/p b−2

naq n 1/p

≥2, 1.11

if the following second-order half-linear diﬀerence equation:

Δ

rn|Δun|^p−2Δun

qn|un 1|^p−2un 1 0 1.12

(3)

has a solutionunsatisfying

ua ub 0, un/≡0, ∀n∈Za, b, 1.13

where and in the sequelq n max{qn,0}.

Applying inequality1.11to1.4 i.e.,1.12withp2, rn 1, andqn≥0, we can obtain the following Lyapunov-type inequality:

b−a^b−2

naqn≥4, 1.14

which was also obtained in5. Whenb−a−1 is odd,1.14is the same as1.6. However, 1.14is worse than1.6whenb−a−1 is even. For more discrete cases and continuous cases for Lyapunov-type inequalities, we refer the reader to5–18.

For a single p-Laplacian equation 1.12, there are many papers which deal with various dynamics behavior of its solutions in the literatures. However, we are not aware of similar works forp-Laplacian systems. We consider here the following quasilinear diﬀerence system of resonant type

−Δ

r1n|Δun|^p¹⁻²Δun

f1n|un 1|^α¹⁻²|vn 1|^α²un 1,

−Δ

r2n|Δvn|^p²⁻²Δvn

f2n|un 1|^β¹|vn 1|^β²⁻²vn 1, 1.15

and the quasilinear diﬀerence system involving thep1, p2, . . . , pn-Laplacian

−Δ

r1n|Δu1n|^p¹⁻²Δu1n

f1n|u1n 1|^α¹⁻²|u2n 1|^α²· · · |umn 1|^α^mu1n 1,

−Δ

r2n|Δu2n|^p²⁻²Δu2n

f2n|u1n 1|^α¹|u2n 1|^α²⁻²· · · |umn 1|^α^mu2n 1, ...

−Δ

rmn|Δumn|^p^m⁻²Δumn

fmn|u1n 1|^α¹|u2n 1|^α²· · · |umn 1|^α^m⁻²umn 1.

1.16

For the sake of convenience, we give the following hypothesesH1 and H2 for system1.15and hypothesisH3for system1.16:

H1r1n, r2n, f1nandf2nare real-valued functions andr1n>0 andr2n>0 for alln∈Z;

H21< p1, p2 <∞,α1, α2, β1, β2>0 satisfyα1/p1 α2/p2 1 andβ1/p1 β2/p21;

H3rin and fin are real-valued functions and rin > 0 for i 1,2, . . . , m.

Furthermore, 1< pi<∞andαi >0 satisfy_m

i1αi/pi 1.

(4)

System1.15and 1.16 are the discrete analogies of the following two quasilinear diﬀerential systems:

−

r1tut^p−2ut

f1t|ut|^α¹⁻²|vt|^α²ut,

−

r2tvt^q−2vt

f2t|ut|^β¹|vt|^β²⁻²vt, 1.17

−

r1tu₁t^p¹⁻²u₁t

f1t|u1t|^α¹⁻²|u2t|^α²· · · |unt|^αⁿu1t,

−

r2tu₂t^p²⁻²u₂t

f2t|u1t|^α¹|u2t|^α²⁻²· · · |unt|^αⁿu2t, ...

−

rntu_nt^pⁿ⁻²u_nt

fnt|u1t|^α¹|u2t|^α²· · · |unt|^αⁿ⁻²unt,

1.18

respectively. Recently, N´apoli and Pinasco 19, Cakmak and Tiryaki 20, 21, and Tang and He 22 established some Lyapunov-type inequalities for systems 1.17 and 1.18.

Motivated by the above-mentioned papers, the purpose of this paper is to establish some Lyapunov-type inequalities for systems1.15and1.16. As a byproduct, we derive a better Lyapunov-type inequality than1.11

b−2 na

⎡

⎢⎣ _n

sars^1/1−p_p−1_b−1

sn 1rs^1/1−p_p−1 _n

sars^1/1−p_p−1 _b−1

sn 1rs^1/1−p_p−1 q n

⎤

⎥⎦≥1 1.19

for the second-order half-linear diﬀerence equation1.12. In particular, 1.19 produces a new Lyapunov-type inequality

1 b−a

b−2 na

n 1−ab−n−1q n≥1 1.20

for Hill’s equation1.4whenp 2 andrt 1. It is easy to see that1.20is better than 1.6.

This paper is organized as follows.Section 2gives some Lyapunov-type inequalities for system 1.15, and Lyapunov-type inequalities for system 1.16 are established in Section 3. InSection 4, we apply our Lyapunov-type inequalities to obtain lower bounds for the first eigencurve in the generalized spectra.

2. Lyapunov-Type Inequalities for System 1.15

In this section, we establish some Lyapunov-type inequalities for system1.15.

(5)

Denote

ζ1n: ⁿ

τa

r1τ^1/1−p¹ _p₁₋₁

, η1n: ^b−1

τn 1

r1τ^1/1−p¹ p1−1

, 2.1

ζ2n: ⁿ

τar2τ^1/1−p² _p₂₋₁

, η2n: ^b−1

τn 1

r2τ^1/1−p² p2−1

. 2.2

Theorem 2.1. Leta, b ∈Zwitha≤b−2. Suppose that hypotheses (H1) and (H2) are satisfied. If system1.15has a solutionun, vnsatisfying the boundary value conditions:

ua ub 0va vb, un/≡0, vn/≡0, ∀n∈Za, b, 2.3

then one has the following inequality:

b−2 na

ζ1nη1n ζ1n η1nf₁n

α1β1/p²₁ b−2 na

ζ1nη1n ζ1n η1nf₂n

β1α2/p1p2

× ^b−2

na

β1α2/p1p2 b−2 na

α2β2/p²₂

≥1,

2.4

where and in the sequelf_in max{fin,0}fori1,2.

Proof. By1.15and2.3, we obtain

b−1 na

r1n|Δun|^p¹^b−2

na

f1n|un 1|^α¹|vn 1|^α², 2.5 b−1

nar2n|Δvn|^p² ^b−2

naf2n|un 1|^β¹|vn 1|^β². 2.6

It follows from2.1,2.3, and the H ¨older inequality that

|un 1|^p¹

n

τaΔuτ

p1

≤ ⁿ

τa|Δuτ| p1

(6)

≤ ⁿ

τar1τ^1/1−p¹

p1−1n

τar1τ|Δuτ|^p¹ ζ1nⁿ

τa

r1τ|Δuτ|^p¹, a≤n≤b−1,

2.7

|un 1|^p¹

b−1 τn 1

Δuτ

p1

≤ ^b−1

τn 1

|Δuτ|

p1

≤ ^b−1

τn 1

r1τ^1/1−p¹

p1−1 b−1 τn 1

r1τ|Δuτ|^p¹

η1n ^b−1

τn 1

r1τ|Δuτ|^p¹, a≤n≤b−1.

2.8

From2.7and2.8, we have

|un 1|^p¹ ≤ ζ1nη1n ζ1n η1n

b−1

τar1τ|Δuτ|^p¹, a≤n≤b−1. 2.9

Now, it follows from2.3,2.5,2.9,H2, and the H ¨older inequality that

b−2

naf₁n|un 1|^p¹≤^b−2

na

_b−1

nar1n|Δun|^p¹ M11

b−2 na

f1n|un 1|^α¹|vn 1|^α²

≤M11

b−2

naf₁n|un 1|^α¹|vn 1|^α²

≤M11

b−2 na

f₁n|un 1|^p¹

α1/p1 b−2 na

f₁n|vn 1|^p² α2/p2

,

2.10

b−2

naf₂n|un 1|^p¹≤^b−2

na

_b−1

nar1n|Δun|^p¹ M12

b−2

naf1n|un 1|^α¹|vn 1|^α²

(7)

≤M12

b−2

naf₁n|un 1|^α¹|vn 1|^α²

≤M12

b−2 na

f₁n|un 1|^p¹

α1/p1 b−2

na

f₁n|vn 1|^p² α2/p2

,

2.11

where

M11^b−2

na

, M12^b−2

na

. 2.12

Similar to the proof of2.9, from2.2and2.3, we have

|vn 1|^p² ≤ ζ2nη2n ζ2n η2n

b−1

τar2τ|Δvτ|^p², a≤n≤b−1. 2.13 It follows from2.3,2.6,2.13,H2, and the H ¨older inequality that

b−2

naf₁n|vn 1|^p² ≤^b−2

na

_b−1

nar2n|Δvn|^p² M21

b−2 na

f2n|un 1|^β¹|vn 1|^β²

≤M21

b−2

naf₂n|un 1|^β¹|vn 1|^β²

≤M21

b−2

naf₂n|un 1|^p¹

β1/p1 b−2

naf₂n|vn 1|^p² β2/p2

, b−2

na

f₂n|vn 1|^p² ≤^b−2

na

_b−1

na

r2n|Δvn|^p²

M22

b−2

naf2n|un 1|^β¹|vn 1|^β²

≤M22

b−2

naf₂n|un 1|^β¹|vn 1|^β²

≤M22

b−2

naf₂n|un 1|^p¹

β1/p1 b−2

naf₂n|vn 1|^p² β2/p2

,

2.14

where

M21^b−2

na

, M22^b−2

na

. 2.15

(8)

Next, we prove that

b−2

naf₁n|un 1|^p¹>0. 2.16

If2.16is not true, then

b−2 na

f₁n|un 1|^p¹0. 2.17

0≤^b−1

nar1n|Δun|^p¹ ^b−2

naf1n|un 1|^α¹|vn 1|^α²≤^b−2

naf₁n|un 1|^α¹|vn 1|^α² 0.

2.18

It follows fromH1that

Δun≡0, a≤n≤b−1. 2.19

Combining2.7with2.19, we obtain thatun≡ 0 fora≤n≤ b, which contradicts2.3.

Therefore,2.16holds. Similarly, we have b−2

na

f₂n|un 1|^p¹>0,

b−2 na

f₁n|vn 1|^p²>0,

b−2 na

f₂n|vn 1|^p²>0. 2.20

From2.10,2.11,2.14,2.16,2.20, andH2, we have

M₁₁^α¹^β¹^/p²¹M₁₂^β¹^α²^/p¹^p²M₂₁^β¹^α²^/p¹^p²M₂₂^α²^β²^/p²² ≥1. 2.21

It follows from2.12,2.15, and2.21that2.4holds.

Corollary 2.2. Leta, b ∈Zwitha≤b−2. Suppose that hypothesis (H1) and (H2) are satisfied. If system1.15has a solutionun, vnsatisfying2.3, then one has the following inequality:

b−2 na

f₁n

ζ1nη1n_1/2α1β1/p²₁ b−2 na

f₂n

ζ1nη1n_1/2β1α2/p1p2

× ^b−2

naf₁n

ζ2nη2n_1/2β1α2/p1p2 b−2 naf₂n

ζ2nη2n_1/2α2β2/p²₂

≥2^p²^β¹^p¹^α²^/p¹^p². 2.22

(9)

Proof. Since

ζin ηin≥2

ζinηin_1/2

, i1,2, 2.23

it follows from2.4andH2that2.22holds.

Corollary 2.3. Leta, b∈Zwitha≤b−2. Suppose that hypotheses (H1) and (H2) are satisfied. If system1.15has a solutionun, vnsatisfying2.3, then one has the following inequality:

b−1

nar1n^1/1−p¹

β1p1−1/p1 b−1

nar2n^1/1−p²

α2p2−1/p2

× ^b−2

naf₁n

β1/p1 b−2 naf₂n

α2/p2

≥2^β¹^α².

2.24

Proof. Since

ζ1nη1n_1/2

ⁿ

τar1τ^1/1−p¹ ^b−1

τn 1

r1τ^1/1−p¹

_p₁_−1/2

≤ 1

2^p¹⁻¹

b−1

τa

r1τ^1/1−p¹ p1−1

, ζ2nη2n1/2 ⁿ

τa

r2τ^1/1−p² ^b−1

τn 1

r2τ^1/1−p²

_p₂_−1/2

≤ 1

2^p²⁻¹

b−1

τar2τ^1/1−p² p2−1

,

2.25

Whenα1 β2 p1 p2 p,α2 β1 0,r1t r2t rt, andf1t f2t qt, system1.15reduces to the second-order half-linear diﬀerence equation 1.12. Hence, we can directly derive the following Lyapunov-type inequality for1.12from2.10and2.16.

Theorem 2.4. Leta, b∈Zwitha≤b−2. Suppose thatp >1 andrn>0. If 1.12has a solution unsatisfying1.13, then one has the following inequality:

b−2 na

⎡

⎢⎣ _n

τarτ^1/1−p_p−1_b−1

τn 1rτ^1/1−p_p−1 _n

τarτ^1/1−p_p−1 _b−1

τn 1rτ^1/1−p_p−1 q n

⎤

⎥⎦≥1. 2.26

(10)

Since

n

τarτ^1/1−p

_p−1 _b−1

τn 1

rτ^1/1−p

p−1

≥2 n

τarτ^1/1−p ^b−1

τn 1

rτ^1/1−p

p−1/2

, 2.27

it follows fromTheorem 2.4that the following corollary holds.

Corollary 2.5. Leta, b∈Zwitha≤b−2. Suppose thatp >1 andrn>0. If1.12has a solution unsatisfying1.13, then one has the following inequality:

b−2 na

⎡

⎣q n ⁿ

τa

rτ^1/1−p ^b−1

τn 1

rτ^1/1−p

_p−1/2⎤

⎦≥2. 2.28

Remark 2.6. It is easy to see that Lyapunov-type inequalities2.26and2.28are better than 1.11.

3. Lyapunov-Type Inequalities for System 1.16

In this section, we establish some Lyapunov-type inequalities for system1.16. Denote

ζin: ⁿ

τariτ^1/1−pⁱ pi−1

, i1,2, . . . , m, 3.1

ηin: ^b−1

τn 1

riτ^1/1−pⁱ pi−1

, i1,2, . . . , m. 3.2

Theorem 3.1. Leta, b∈Zwitha≤b−2. Suppose that hypothesis (H3) is satisfied. If system1.16 has a solutionu1n, u2n, . . . , umnsatisfying the boundary value conditions:

uia uib 0, uin/≡0, ∀n∈Za, b, i1,2, . . . , m, 3.3

then one has the following inequality:

m i1

m j1

b−2 na

ζinηin ζin ηinf_jn

^αⁱ^α^j^/pⁱ^p^j

≥1. 3.4

Proof. By1.16,H3, and3.3, we obtain b−1

na

rin|Δuin|^pⁱ^b−2

na

fin^m

k1

|ukn 1|^α^k

, i1,2, . . . , m. 3.5

(11)

It follows from3.1,3.3, and the H ¨older inequality that

|uin 1|^pⁱ

n τa

Δuiτ

pi

≤ ⁿ

τa

riτ^1/1−pⁱ

_p_i₋₁_n

τa

riτ|Δuiτ|^pⁱ

ζinⁿ

τariτ|Δuiτ|^pⁱ, a≤n≤b−1, i1,2, . . . , m.

3.6

Similarly, it follows from3.2,3.3, and the H ¨older inequality that

|uin 1|^pⁱ

b−1 τn 1

Δuiτ

pi

≤ ^b−1

τn 1

riτ^1/1−pⁱ

pi−1 b−1 τn 1

riτ|Δuiτ|^pⁱ

ηin^b−1

τn 1

riτ|Δuiτ|^pⁱ, a≤n≤b−1, i1,2, . . . , m.

3.7

|uin 1|^pⁱ≤ ζinηin ζin ηin

b−1 τa

riτ|Δuiτ|^pⁱ, a≤n≤b−1, i1,2, . . . , m. 3.8

Now, it follows from3.3,3.5,3.8,H3, and the generalized H ¨older inequality that

b−2 na

f_jn|uin 1|^pⁱ ≤^b−2

na

_b−1

τa

riτ|Δuiτ|^pⁱ

Mij

b−2 na

fin^m

k1

|ukn 1|^α^k

≤Mij

b−2 na

f_in^m

k1

|ukn 1|^α^k

≤Mij

m k1

_b−2

naf_i n|ukn 1|^p^k αk/pk

,

3.9

where

Mij^b−2

na

, i, j 1,2, . . . , m. 3.10

(12)

Next, we prove that b−2

naf_i n|ukn 1|^p^k >0, i, k1,2, . . . , m. 3.11

If3.11is not true, then there existsi0, k0∈ {1,2, . . . , m}such that b−2

naf_i₀n|uk0n 1|^p^k⁰ 0. 3.12

From3.5,3.12, and the generalized H ¨older inequality, we have

0≤^b−1

na

ri0n|Δui0n|^pⁱ⁰ ^b−2

na

fi0n^m

k1

|ukn 1|^α^k ≤^m

k1

_b−2

na

f_i₀n|ukn 1|^p^k αk/pk

0.

3.13

It follows from the fact thatri0n>0 that

Δui0n≡0, a≤n≤b−1. 3.14

Combining3.6with3.14, we obtain thatui0n≡0 fora≤n≤b, which contradicts3.3.

Therefore,3.11holds. From3.9,3.11, andH3, we have m

i1

m j1

M^α_ijⁱ^α^j^/pⁱ^p^j≥1. 3.15

It follows from3.10and3.15that3.4holds.

Corollary 3.2. Leta, b∈Zwitha≤b−2. Suppose that hypothesis (H3) is satisfied. If system1.16 has a solutionu1n, u2n, . . . , umnsatisfying3.3, then one has the following inequality:

m i1

m j1

b−2 naf_jn

ζinηin_1/2αiαj/pipj

≥2. 3.16

Proof. Since

ζin ηin≥2

ζinηin1/2

, i1,2, . . . , m, 3.17

(13)

Corollary 3.3. Leta, b∈Zwitha≤b−2. Suppose that hypothesis (H3) is satisfied. If system1.16 has a solutionu1n, u2n, . . . , umnsatisfying3.3, then one has the following inequality

m i1

b−1 na

rin^1/1−pⁱ

αipi−1/pim j1

b−2 naf_jn

αj/pj

≥2^A, 3.18

whereA_m

i1αi. Proof. Since

ζinηin1/2 ⁿ

τa

rin^1/1−pⁱ ^b−1

τn 1

riτ^1/1−pⁱ

_p_i_−1/2

≤ 1

2^pⁱ⁻¹ b−1

τariτ^1/1−pⁱ pi−1

, i1,2, . . . , m,

3.19

4. Some Applications

In this section, we apply our Lyapunov-type inequalities to obtain lower bounds for the first eigencurve in the generalized spectra.

Leta, b ∈ Zwitha ≤ b−2. We consider here a quasilinear diﬀerence system of the form:

−Δ

|Δu1n|^p¹⁻²Δu1n

λ1α1qn|u1n 1|^α¹⁻²|u2n 1|^α²· · · |umn 1|^α^mu1n 1,

−Δ

|Δu2n|^p²⁻²Δu2n

λ2α2qn|u1n 1|^α¹|u2n 1|^α²⁻²· · · |umn 1|^α^mu2n 1, ...

−Δ

|Δumn|^p^m⁻²Δumn

λmαmqn|u1n 1|^α¹|u2n 1|^α²· · · |umn 1|^α^m⁻²umn 1,

4.1

whereqn > 0,λi ∈ R,pi and αi are the same as those in1.16, andui satisfies Dirichlet boundary conditions:

uia uib 0, uin>0, ∀n∈Za 1, b−1, i1,2, . . . , m. 4.2 We define the generalized spectrum S of a nonlinear diﬀerence system as the set of vectorλ1, λ2, . . . , λm ∈ R^m such that the eigenvalue problem4.1with 4.2admits a nontrivial solution.

(14)

Eigenvalue problem or boundary value problem4.1with4.2is a generalization of the followingp-Laplacian diﬀerence equation

−Δ

|Δun|^p−2Δun

λpqn|un 1|^p−2un 1, 4.3

with Dirichlet boundary condition:

ua 0ub, un>0, ∀n∈Za 1, b−1, 4.4

where p > 1, λ ∈ R, and qn > 0. When p 2, Atkinson 1, Theorems 4.3.1 and 4.3.5 investigated the existence of eigenvalues for4.3with4.4, see also23.

Letfin λiαiqnandrin 1 fori1,2, . . . , m. Then we can applyTheorem 3.1to boundary value problem4.1with4.2and obtain a lower bound for the first eigencurve in the generalized spectra.

Theorem 4.1. Leta, b∈Zwitha≤b−2. Assume that 1 < pi<∞,αi >0 satisfy_m

i1α_i/pi

1, and that qn > 0 for alln ∈ Z. Then there exists a functionhλ1, . . . , λ_m−1 such thatλm ≥ hλ1, . . . , λm−1for every generalized eigenvalueλ1, λ2, . . .,λmof boundary value problem4.1with 4.2, wherehλ1, . . . , λm−1is given by:

hλ1, . . . , λm−1

1 αm

⎡

⎣^m−1

j1

λjαj

_α_j_/p_j^m

i1

b−2 na

n−a 1b−n−1^pⁱ⁻¹ n−a 1^pⁱ⁻¹ b−n−1^pⁱ⁻¹qn

αi/pi⎤

⎦

−p_m/αm

. 4.5

Proof. For the eigenvalue λ1, λ2, . . . , λm, 4.1 with 4.2 has a nontrivial solution u1n, u2n, . . . , umn. That is system1.16 with rin 1 and fin λiαiqn has a solutionu1n, u2n, . . . , umnsatisfying3.3, it follows from3.4thatfin λiαiqn>

0,for alln∈Z,i1,2, . . . , m, and that

1≤^m

i1

m j1

b−2

na

αiαj/pipj

^m

j1

λjαj

αj/pj

m i1

b−2 na

ζinηin ζin ηinqn

αi/pi

^m

j1

λjαj

_α_j_/p_j^m

i1

b−2 na

αi/pi

.

4.6

Hence, we have

λm≥ 1 αm

⎡

⎣^m−1

j1

λjαj

αj/pj

m i1

b−2 na

αi/pi⎤

⎦

−pm/αm

. 4.7

This completes the proof ofTheorem 4.1.

(15)

Whenm2, boundary value problem4.1with4.2reduces to the simpler form:

−Δ

|Δu1n|^p¹⁻²Δu1n

λ1α1qn|u1n 1|^α¹⁻²|u2n 1|^α²u1n 1,

−Δ

|Δu2n|^p²⁻²Δu2n

λ2α2qn|u1n 1|^α¹|u2n 1|^α²⁻²u2n 1, 4.8

with Dirichlet boundary conditions:

uia uib 0, uin>0, ∀n∈Za 1, b−1, i1,2, 4.9

where 1< p1, p2<∞,α1, α2 >0 satisfyα1/p1 α2/p21, andqn>0 for alln∈Z.

Applying Theorem 4.1 to system 4.8 with 4.9 and system 4.3 with 4.4, respectively, we have the following two corollaries immediately.

Corollary 4.2. Leta, b ∈ Zwith a ≤ b−2. Assume that 1 < p1, p2 < ∞, α1, α2 > 0 satisfy α1/p1 α2/p2 1, and thatqn > 0 for alln ∈ Z. Then there exists a functionhλ1such that λ2 ≥hλ1for every generalized eigenvalueλ1, λ2of system4.8with4.9, wherehλ1is given by:

hλ1

1/α2

λ1α1

_b−2

na

XY^p¹⁻¹/

X^p¹⁻¹ Y^p¹⁻¹ qn_p

2/α2−1_b−2

na

XY^p²⁻¹/

X^p²⁻¹ Y^p²⁻¹ qn, 4.10

whereXdenoten−a 1andYdenoteb−n−1.

Corollary 4.3. Leta, b∈Zwitha≤b−2. Assume thatp >1 andqn>0 for alln∈Z. Then for every eigenvalueλof system4.3with4.4, one has

λ≥ 1 p

_b−2

na

n−a 1b−n−1^p−1 n−a 1^p−1 b−n−1^p−1qn

₋₁

. 4.11

References

1 F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, NY, USA, 1964.

2 S. S. Cheng, “A discrete analogue of the inequality of Lyapunov,” Hokkaido Mathematical Journal, vol.

12, no. 1, pp. 105–112, 1983.

3 A. M. Liapounoff, “Probléme général de la stabilité du mouvement,” Annales de la Faculté des Sciences de Toulouse, vol. 2, pp. 203–407, 1907.

4 M. ¨Unal, D. Cakmak, and A. Tiryaki, “A discrete analogue of Lyapunov-type inequalities for non- linear systems,” Computers & Mathematics with Applications, vol. 55, no. 11, pp. 2631–2642, 2008.

5 G. Sh. Guseinov and B. Kaymakcalan, “Lyapunov inequalities for discrete linear Hamiltonian systems,” Computers & Mathematics with Applications, vol. 45, pp. 1399–1416, 2003.

6 S. S. Cheng and T. T. Lu, “Convex regular domains of tridiagonal matrices,” Linear Algebra and its Applications, vol. 79, pp. 103–125, 1986.

7 S. S. Cheng and L. Y. Hsieh, “On discrete analogue of Lyapunov’s inequality: addendum,” Tamkang Journal of Mathematics, vol. 20, no. 4, pp. 333–339, 1989.

(16)

8 S. S. Cheng, “A sharp condition for the ground state of diﬀerence equation,” Applied Analytical Analyt- ical, vol. 34, no. 2, pp. 105–109, 1989.

9 S. S. Cheng, L. Y. Hsieh, and D. Chao, “Discrete Lyapunov inequality conditions for partial diﬀerence equations,” Hokkaido Mathematical Journal, vol. 19, no. 2, pp. 229–239, 1990.

10 S. S. Cheng, “Lyapunov inequality for system disconjugacy of even order diﬀerential equations,”

Tamkang Journal of Mathematics, vol. 22, no. 2, pp. 193–197, 1991.

11 S. S. Cheng, “Lyapunov inequalities for diﬀerential and diﬀerence equations,” Fasiculi Mathematici, no.

23, pp. 25–41, 1991.

12 S. Clark and D. B. Hinton, “Discrete Lyapunov inequalities for linear Hamiltonian systems,” Mathe- matical Inequalities & Applications, vol. 1, no. 2, pp. 201–209, 1998.

13 S. Clark and D. B. Hinton, “Discrete Lyapunov inequalities,” Dynamic Systems and Applications, vol. 8, no. 3-4, pp. 369–380, 1999.

14 Z. He, “Existence of two solutions of m-point boundary value problem for second order dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 97–109, 2004.

15 L. Q. Jiang and Z. Zhou, “Lyapunov inequality for linear Hamiltonian systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 310, no. 2, pp. 579–593, 2005.

16 S. H. Lin and G. S. Yang, “On discrete analogue of Lyapunov’s inequality,” Tamkang Journal of Mathe- matics, vol. 20, no. 2, pp. 169–186, 1989.

17 X. Wang, “Stability criteria for linear periodic Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 329–336, 2010.

18 Q. Zhang and X. H. Tang, “Lyapunov inequalities and stability for discrete linear Hamiltonian systems,” Applied Mathematics and Computation, vol. 218, no. 2, pp. 574–582, 2011.

19 P. L. N´apoli and J. P. Pinasco, “Estimates for eigenvalues of quasilinear elliptic systems,” Journal of Dif- ferential Equations, vol. 227, no. 1, pp. 102–115, 2006.

20 D. Cakmak and A. Tiryaki, “On Lyapunov-type inequality for quasilinear systems,” Applied Mathe- matics and Computation, vol. 216, no. 12, pp. 3584–3591, 2010.

21 D. Cakmak and A. Tiryaki, “Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving thep1, p2, ..., pn-Laplacian,,” Journal of Mathematical Analysis and Applications, vol. 369, no.

1, pp. 76–81, 2010.

22 X. H. Tang and X. He, “Lower bounds for generalized eigenvalues of the quasilinear systems,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 72–85, 2012.

23 Y. Wang and Y. Shi, “Eigenvalues of second-order diﬀerence equations with periodic and antiperiodic boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 309, no. 1, pp. 56–69, 2005.

(17)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Probability and Statistics

Journal of

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Operations Research

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Algebra

Discrete Dynamics in Nature and Society

Decision Sciences

Discrete Mathematics

^{Journal of}

Lyapunov-Type Inequalities for the Quasilinear Difference Systems

Research Article