Existence of Multiple Solutions for a Class of n Dimensional Discrete Boundary Value Problems

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Volume 2010, Article ID 198465,14pages doi:10.1155/2010/198465

Research Article

Existence of Multiple Solutions for a Class of

n

-Dimensional Discrete Boundary Value Problems

Weiming Tan

1, 2

_{and Zhan Zhou}

1

1_{College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China} 2_{Institute of Mathematics and Physics, Wuzhou University, Wuzhou 543002, China}

Correspondence should be addressed to Zhan Zhou,[email protected]

Received 21 July 2009; Accepted 20 January 2010

Academic Editor: Raul F. Manasevich

Copyrightq2010 W. Tan and Z. Zhou. 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 using critical point theory, we obtain some new results on the existence of multiple solutions for a class ofn-dimensional discrete boundary value problems. Results obtained extend or improve existing ones.

1. Introduction

LetN,Z,Rbe the set of all natural numbers, integers, and real numbers, respectively. For any

a, b∈Z, defineZa {a, a1, a2, . . .},Za, b {a, a1, . . . , b}whena < b.

In this paper, we consider the existence of multiple solutions for the following n -dimensional discrete nonlinear boundary value problem:

ΔpkΔXk−1qkXk fk, Xk, k∈Za1, b,

Xa αXa1 A, Xb1 βXb B, 1.1

where n ∈ N, Xk x1k, x2k, . . . , xnkT ∈ Rn, Δ is the forward diﬀerence

operator defined by ΔXk Xk 1 − Xk, f : Z × Rn _→ _Rn _with _f_{k, U}

f1k, U, f2k, U, . . . , fnk, UT, and is continuous forU.a, b ∈Zwitha < b,αandβare

constants,A a1, a2, . . . , anT andB b1, b2, . . . , bnT aren-dimensional vectors,pkand

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that the uniqueness of solutions implies the existence of solutions for some conjugate boundary value problems. In recent years, by using various methods and techniques, such as nonlinear alternative of Leray-Schauder type, the cone theoretic fixed point theorem, and the method of upper and lower solution, a series of existence results for the solutions of the BVP 1.1 in some special cases, for example, n 1, α β 0, A B 0;

n1, α0, β−1, AB0 have been obtained in literatures. We refer to3–7 .

The critical point theory has been an important tool for investigating the periodic solutions and boundary value problems of diﬀerential equations8–10 . In recent years, it is applied to the study of periodic solutions11–13 and boundary value problems14–17 of diﬀerence equations.

For the case whenn 1, the scalar BVP 1.1 was studied by Yu and Guo in17 . By using critical point theorem, they obtained various conditions to guarantee the existence of one solution, but they did not obtain the existence conditions of multiple solutions. In this scalar case, the BVP can be viewed as the discrete analogue of the following self-adjoint diﬀerential equation:

d dt

ptdy

dt

qtyft, y, 1.2

which is a generalization of Emden-Fowler equation:

d dt

tρdy

dt

tδyγ0. 1.3

The Emden-Fowler equation originated from earlier theories of gaseous dynamics in astrophysics 18 , and later, found applications in the study of fluid mechanics, relative mechanics, nuclear physics, and in the study of chemical reaction system19 .

For the case wherepk ≡ −1,qk ≡0,A B θthe zero vector ofRn, BVP1.1

are reduced to

−Δ2_X_k₋₁ _f_{k, X}_k_, _k_∈_Z_a₁_{, b}_,

Xa −αXa1, Xb1 −βXb,

1.4

which were studied by Jiang and Zhou in15 . They obtained the existence results of multiple solutions by using critical point theory again.

In the aforementioned references, most of the diﬀerence equations involved are scalar. The purpose of this paper is further to demonstrate the powerfulness of critical point theory in the study of existence of discrete boundary value problems and obtain various conditions for the existence of at least two nontrivial solutions for the BVP1.1.

The remaining of this paper is organized as follows. First, inSection 2, we will establish the variational framework associated with1.1and transfer the problem of the existence of solutions of1.1into that of the existence of critical points of the corresponding functional. Some basic results will also be recalled. Then, inSection 3, we present various new conditions on the existence of at least two nontrivial solutions for the BVP 1.1. Some examples are given to illustrate the conclusions. We mention that our results generalize the ones in15

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To conclude the introduction, we refer to 21, 22 for the general background on diﬀerence equations.

2. Preliminary and Variational Framework

LetEbe a real Hilbert space,J ∈ C1_E,_R_{, which means that}_J _{is a continuously}

Fre´chet-diﬀerentiable functional defined on E. J is said to satisfy Palais-Smale condition P-S condition for short, if any sequence {xn}∞n1 ⊂ E for which {Jxn}∞n1 is bounded and

Jxn → 0 asn → ∞possesses a convergent subsequence inE.

LetBρ be the open ball inE with radiusρ and centered at 0 and let∂Bρ denote its

boundary. The following lemmas will be useful in the proofs of our main results.

Lemma 2.1. (Mountain Pass Lemma [10]). Let E be a real Hilbert space, and assume that J ∈ C1_E,_R_{satisfies the P-S condition and the following conditions.}

1There exist constantρ >0 anda >0 such thatJx≥a,∀x∈∂Bρ, whereBρ{x∈E: xE< ρ}.

2J0≤0 and there existsx0/∈Bρsuch thatJx0≤0.

ThenJpossesses a critical valuec≥a. Moreover,cinfh∈Γmaxs∈0,1Jhs,where

Γ {h∈C0,1 , E|h0 0, h1 x0}. 2.1

Lemma 2.2. Letai≥0,γ >2,i∈Z1, m, then

1

√

m

i1

ai≤

m

i1

a2

i ≤ m

i1

ai. 2.2

Without loss of generality, we assume that a 0,b mwhere m ∈ N, and there exists a function Pk, U ∈ C1_Z₁_{, m}_×_Rn_,_R_{, such that for any} _{k, U} _∈ _Z₁_{, m}_×_Rn_, ∇UPk, U fk, U, where∇UPk, U ∂P/∂u1, . . . , ∂P/∂unT is the gradient ofPk, U

inU, andPk, θ≥0 whereθis the zero vector inRn_.

Letck qk−pk−pk1, then BVP1.1reduces to

pk1Xk1 ckXk pkXk−1 fk, Xk, k∈Z1, m, 2.3

X0 αX1 A, Xm1 βXm B, 2.4

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Define

HX X1, . . . , XmT |Xk x1k, . . . , xnk∈Rn,∀k∈Z1, m

. 2.5

Then,Hcan be equipped with the inner product

X, YH m

k1

Xk, Yk

m

k1

n

i1

xikyik, 2.6

by which the norm · Hcan be induced by

XH

m

k1

Xk2n, ∀X∈H, 2.7

whereXk x1k, x2k, . . . , xnkT,Yk y1k, y2k, . . . , ynkT ∈Rn, and · nand

·,·are the norm and the inner product inRn_{, respectively.}

Define a linear mappingL:H → Rmn_by

LX x11, . . . , x1m, x21, . . . , x2m, . . . , xn1, . . . , xnmT. 2.8

ThenLis a linear and one to one mapping. Clearly,LXmnXH.

With this mapping, BVP2.3and2.4can be represented by the matrix equation:

MLX NLFX, 2.9

whereX∈H,FX f1, X1, f2, X2, . . . , fm, Xm∈Hand

M ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Q · · · 0

Q · · ·

..

. ... . .. ...

0 · · · Q ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

mn×mn

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with Q ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

c1−αp1 p2 0 · · · 0 0

p2 c2 p3 · · · 0 0 0 p3 c3 · · · 0 0

· · · ·

0 0 0 · · · cm−1 pm

0 0 0 · · · pm cm−βpm1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

m×m

, N ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ η1 η2 .. . ηn ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

mn×1

, ηj

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

p1aj

0 .. .

0

pm1bj

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

m×1

, j1,2, . . . , n.

2.11

Define a functionalJonHas

JX 1

2MLX, LX N, LX−WX, 2.12

whereWX m_k₁Pk, Xk. ThenJ∈C1_H,_R_.

For anyX X1, X2, . . . , Xm∈H, denote

X X0, X1, . . . , Xm, Xm1, 2.13

whereX0, X1, Xm, Xm1satisfying2.4. Thus, there is a one to one correspondence fromHto

HX0, X1, . . . , Xm1|X0 αX1 A, Xm1 βXm B. 2.14

Clearly,JX 0 if and only ifXsatisfies2.3and2.4. Therefore, the existence of solutions to BVP2.3and2.4is transferred to the existence of the critical point of the functionalJon

H.

By a solution{Xk}m_k₀1 of2.3and2.4, we mean that{Xk}m_k₁satisfies2.3, and

2.4holds.{Xk}m_k₀1is nontrivial ifX1, . . . , XmH/0.

3. Main Results

In this section, we will suppose that the matrixQdefined in2.11is positive definite,λmin,

λmax are the minimal eigenvalue and maximal eigenvalue ofQ, respectively. It is clear that

λmin,λmaxare also the minimal eigenvalue and maximal eigenvalue ofM, respectively. The

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Theorem 3.1. If there exist constantsα1,α2 : 0 < α1 < λmin ≤ λmax < α2, and ρ > 0,γ ≥ 0,

δ > d2Nmn/λ_min−α1satisfying

WX≤ 1

2α1X

2

H, ∀X∈H,XH< δ, 3.1

WX≥ 1

2α2X

2

H−γ, ∀X∈H,XH> ρ, 3.2

then BVP2.3and2.4has at least two nontrivial solutions.

Proof. SincePk, θ≥ 0, we know thatWθ≥ 0 whereθdenotes the zero element ofH. By

3.1, we getWθ 0 andJθ 0. Let

ωmaxWX−1

2α2X

2

Hγ

:X∈H,XH≤ρ

. 3.3

Then by3.2, we get

WX≥ 1

2α2X

2

H−

γω 1

2α2X

2

H−γ, ∀X∈H, 3.4

whereγγω≥0. Then for all X∈H, we have

JX 1

2MLX, LX N, LX−WX

≤ 1

2λmaxX

2

HNmnXH−

1 2α2X

2

H−γ

λmax−α2

2 X

2

HNmnXHγ.

3.5

Sinceλmax< α2, we see thatJX → −∞asXH → ∞. ThusJXis bounded from

above onH, andJXcan achieve its maximum onH. In other words, there existsX1 ∈H,

such thatJX1 supX∈HJX. SoX1is a critical point ofJX, andX1is a solution of BVP

2.3and2.4.

For anyX∈Hwithd <XH < δ, we have

JX≥ λmin−α1

2 X

2

H− NmnXH>0. 3.6

SoJX1 supX∈HJX>0 andX1is a nontrivial solution to BVP2.3and2.4.

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In fact, for any sequence{Xk_}∞

k1inH,{JXk}∞k1is bounded and limk→ ∞JXk

0, then there existsK >0 such thatJXk_>₋_K_{, and it follows from}_3.5_that

−K≤JXk≤ λmax−α2

2

Xk2

HNmn

Xk

Hγ

_, _3.7

that is,

λmax−α2

2 Xk

2

HNmnXkHγK≥0. 3.8

Since λmax < α2, this implies that {Xk}∞k1 is bounded and possesses a convergent

subsequence. SoJXsatisfies P-S condition onH.

Choosingr∈d, δ, then forX ∈∂Br {X∈H:XHr}, from3.6, we get

JX≥ λmin−α1

2 r

2₋_N

mnr a >0. 3.9

This shows thatJsatisfies condition1of the Mountain Pass Lemma.

On the other hand, from3.5,JX → −∞ asXH → ∞, so there existsP > r

such thatJX <0 forXH > P. PickX0 ∈Hsuch thatX0H > P > r, thenX0 ∈H\Br,

andJX0 < 0. So the condition2of the Mountain Pass Lemma is satisfied. Therefore,J

possesses a critical valuecinfh∈Γmaxs∈0,1Jhs,where

Γ h∈C0,1 , H:h0 θ, h1 X0

. 3.10

A critical point corresponding to c is nontrivial as c ≥ a > 0. LetX2 be a critical point

corresponding to the critical valuecofJ. IfX2/X1, then we are done. Otherwise,X2 X1,

which gives

max

X∈HJX JX1 JX2 infh∈Γsmax∈0,1 Jhs. 3.11

Pickhs sX0, s∈0,1 , thenh∈Γ, and we have

max

X∈HJX JX1 JX2 infh∈Γsmax∈0,1Jhs≤smax∈0,1 JsX0. 3.12

Thus, there existss0 ∈0,1such thatJX2 Js0X0, andX3 s0X0is also a critical point

ofJXinH.

IfX3/X1, thenTheorem 3.1holds. Otherwise,X3 X1 s0X0. In this situation, we

replace X0 with−X0 in the above arguments; thenJ possesses a critical valuec∗ ≥ aand

c∗infh∈Γ∗max_s_∈₀_,₁Jhs,where

Γ∗_h_∈_C₀_,₁ _{, H}_:_h₀ _{θ, h}₁ ₋_X

0

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Assume thatX4is a critical point corresponding toc∗. IfX4/X1, then the proof is complete.

Otherwise,X4X1. Similarly, we can find a critical pointX5ofJsuch thatX5−s1X0holds

for somes1 ∈0,1. Clearly,X5/X3. The proof ofTheorem 3.1is now complete.

FromTheorem 3.1, we have the following corollaries.

Corollary 3.2. If there exist constantsα1,α2 : 0 < α1 < λmin ≤ λmax < α2, andρ > 0, γ ≥ 0,

δ > d2Nmn/λmin−α1satisfying

Pk, U≤ 1

2α1U

2

n, Un< δ, k∈Z1, m,

Pk, U≥ 1

2α2U

2

n−γ, Un> ρ, k∈Z1, m.

3.14

Then BVP2.3and2.4has at least two nontrivial solutions.

Proof. For anyX∈H,XH< δ, thenXkn≤ XH< δ,k∈Z1, m. So

WX

m

k1

Pk, Xk≤

m

k1

1

2α1Xk

2

n

1 2α1X

2

H, 3.15

and condition3.1ofTheorem 3.1is satisfied. Let

γmaxPk, U−1

2α2U

2

nγ

:Un≤ρ, k∈Z1, m

. 3.16

Then

Pk, U≥ 1

2α2U

2

n−γ1, k∈Z1, m, U∈Rn, 3.17

whereγ1 γγ. So, for anyX∈H,

WX

m

k1

Pk, Xk≥

m

k1

1 2

α2Xk2n−γ1

1

2α2X

2

H−mγ1, 3.18

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Remark 3.3. For the special case wherepk≡ −1, qk≡0, ABθ, the BVP2.3and2.4

was studied in15 . Here, the corresponding matrixQbecomes

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

2α −1 0 . . . 0 0

−1 2 −1 . . . 0 0 0 −1 2 . . . 0 0

. . . . . . . . . .

0 0 0 . . . 2 −1

0 0 0 . . . −1 2β ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

m×m

, 3.19

which is positive definite whenα > −1, β ≥ −1 andN 0, . . . ,0T. So,d 0. In this case,

Corollary 3.2reduces to Theorem 3.1 of15 . Therefore, our results extend the ones in15 .

Corollary 3.2also improves the conclusion of Theorem 1 in20 .

Corollary 3.4. If there exist constantsα1< λmin,α2>0,ρ >0,γ >0,δ > d2Nmn/λmin−α1

andτ >2 such that3.1holds and

WX≥ 1

2α2X

τ

H−γ, ∀X∈H,XH> ρ, 3.20

Proof. It suﬃces to prove that 3.20 implies 3.2. In fact, pick ρ max{ρ,λmax

1/α21/τ−2}, then

WX≥ 1

2α2X

τ−2

H · X2H−γ≥

1

2λmax1X

2

H−γ 3.21

forXH> ρ, so condition3.2ofTheorem 3.1is satisfied, and the proof are complete.

The following corollaries is obvious.

Corollary 3.5. If there existα1: 0< α1< λmin,δ > d2Nmn/λmin−α1such that3.1holds

and

lim

XH→ ∞ WX

X2H

β > λmax

2 . 3.22

Corollary 3.6. If there exists constantsα1: 0< α1 < λmin,δ > d2Nmn/λmin−α1such that

3.14holds and

lim

Un→ ∞

Pk, U

U2n

> λmax

2 , ∀k∈Z1, m. 3.23

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Now we will give an example to illustrate our results.

Example 3.7. Consider the discrete boundary problem:

2Xk1 10Xk 2Xk−1 fk, Xk, k∈Z1,5 3.24

with

X0 X1 A, X6 2X5 B, 3.25

where

fk, U 9

400kU

4U22−2 sinU22

3.26

fork∈Z1,5andA 1,0T,B 0,0T.

In this example,m 5, n 2,pk ≡ 2, ck ≡ 10,α 1,β 2,η1 2,0,0,0,0T,

η2 0,0,0,0,0T,N 2,0,0,0,0,0,0,0,0,0T,

M

Q

Q , Q ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

8 2 0 0 0

2 10 2 0 0

0 2 10 2 0

0 0 2 10 2

0 0 0 2 6

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,

Pk, U 9

400k

U42−

1−cosU22

.

3.27

So,N10 2. With the Matlab software, we can get the approximate eigenvalue of matrix Q:

λ14.9957, λ2 6.3645, λ3 8.4127, λ411.0445, λ513.1826. 3.28

ThusQis positive definite and 4.9< λmin <5, 13.1< λmax<13.2. Takeα1 4.5, we find that

8 4

5−4.5 < d

2N10

λmin−α1 <

4

4.9−4.5 10. 3.29

Pickα214,δ10> d,ρ100, then for anyU∈R2:U2< δ10,k∈Z1,5, we have

Pk, U≤ 9

400U

4 2<

9 4U

2 2

α1

2 U

2

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For anyU∈R2_,_U

2> ρ100,k∈Z1,5,

Pk, U 9

400k

U4₂−1−cosU2₂≥ 9

2000

U4₂−2> α2

2 U

2 2−

9

1000. 3.31

In view ofCorollary 3.2, we see that the boundary value problem3.24and3.25has at least two nontrivial solutions.

By a similar method, we can obtain the following results.

Theorem 3.8. Assume thatPk, θ 0 for anyk ∈ Z1, m. If there exist constantsα1,α2 : 0 <

α2< λmin ≤λmax< α1,ρ >0,γ≥0, andδ > d2Nmn/α1−λmaxsatisfying

WX≥ 1

2α1X

2

H, ∀X∈H,XH< δ, 3.32

WX≤ 1

2α2X

2

Hγ, ∀X∈H,XH> ρ. 3.33

Corollary 3.9. Assume thatPk, θ 0 for anyk ∈Z1, m. If there exist constantsα1,α2 : 0 <

α2< λmin ≤λmax< α1,ρ >0,γ≥0, andδ > d2Nmn/α1−λmaxsatisfying

Pk, U≥ 1

2α1U

2

n, Un< δ, k∈Z1, m, 3.34

Pk, U≤ 1

2α2U

2

nγ, Un> ρ, k∈Z1, m, 3.35

Corollary 3.10. Assume thatPk, θ 0 for anyk ∈Z1, m. If there exist constantsα1 > λmax,

α2>0,δ > d2Nmn/α1−λmaxand 1< τ <2 such that3.32holds and

WX≤ 1

2α2X

τ

Hγ, ∀X∈H,XH> ρ, 3.36

Corollary 3.11. Assume thatPk, θ 0 for anyk ∈Z1, m. If there exist constantsα1 > λmax,

andδ > d2Nmn/α1−λmaxsuch that3.32holds and

lim

XH→ ∞ WX

XH

β < λmin

2 , 3.37

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Corollary 3.12. Assume thatPk, θ 0 for anyk∈Z1, m. If there existα1> λmax, andδ > d

2Nmn/α1−λmaxsuch that3.34holds and

lim

Un→ ∞

Pk, U

U2n

< λmin

2 , k∈Z1, m, 3.38

At last, we will give another example.

Example 3.13. Consider the boundary value problem3.24and3.25wherefk, uis replaced by

fk, U

8004U2

k2U22

2 sinU2₂ U. 3.39

Clearly,

Pk, U 800U

2 2

k2U2

1−cosU22

, 3.40

andPk, θ 0. Pickα113.4,α2 4.8, then we have 0< α2 < λmin≤λmax< α1, and

40 3

4

13.4−13.1 < d

2Nmn

α1−λmax <

4

13.4−13.2 20. 3.41

Pickδ20> d,ρ200, then for anyU∈R2_:_U

2 < δ20, we can see that

Pk, U 800U

2 2

k2U2

1−cosU22

≥ 800U22

5220

80U22

11 >

α1

2 U

2

2. 3.42

For anyU∈R2_:_U

2> ρ200, we can get

Pk, U 800U

2 2

k2U2

1−cosU22

≤ 800U22

2200 2

80U22

202 2<

α2

2 U

2

22. 3.43

According toCorollary 3.9, we know that the given BVP in this example has at least two nontrivial solutions.

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Acknowledgments

This work is supported by the Specialized Fund for the Doctoral Program of Higher Eduction.

no. 20071078001, by the Natural Science Foundation of GuangXi no. 0991279, by the Foundation of Education Department of GuangXi Province no. 200807MS121, and by the project of Scientific Research Innovation Academic Group for the Education System of Guangzhou City.

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