Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 345916, 10 pages
doi:10.1155/2008/345916
Research Article
Three Solutions to Dirichlet Boundary Value
Problems for p-Laplacian Difference Equations
Liqun Jiang1, 2 and Zhan Zhou2, 3
1
Department of Mathematics and Computer Science, Jishou University, Jishou, Hunan 416000, China
Department of Applied Mathematics, Hunan University, Changsha, Hunan 410082, China
3
Department of Applied Mathematics, Guangzhou University, Guangzhou 510006, Guangdong, China
2
Correspondence should be addressed to Liqun Jiang, liqunjianghn@yahoo.com
Received 2 March 2007; Revised 16 July 2007; Accepted 15 October 2007
Recommended by Svatoslav Stanek
We deal with Dirichlet boundary value problems for p-Laplacian difference equations depending
on a parameter λ. Under some assumptions, we verify the existence of at least three solutions when
λ lies in two exactly determined open intervals respectively. Moreover, the norms of these solutions
are uniformly bounded in respect to λ belonging to one of the two open intervals.
Copyright q 2008 L. Jiang 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.
1. Introduction
Let R, Z, N be all real numbers, integers, and positive integers, respectively. Denote Za {a, a 1, . . . } and Za, b {a, a 1, . . . , b} with a < b for any a, b ∈ Z.
In this paper, we consider the following discrete Dirichlet boundary value problems:
Δ φp Δxk − 1 λfk, xk 0,
x0 0 xT 1,
k ∈ Z1, T ,
1.1
where T is a positive integer, p > 1 is a constant, Δ is the forward difference operator defined by
Δxk xk 1 − xk, φp s is a p-Laplacian operator, that is, φp s |s|p−2 s, fk, · ∈ CR, R
for any k ∈ Z1, T .
There seems to be increasing interest in the existence of solutions to boundary value
problems for finite difference equations with p-Laplacian operator, because of their applications in many fields. Results on this topic are usually achieved by using various fixed point
theorems in cone; see 1–4 and references therein for details. It is well known that critical point theory is an important tool to deal with the problems for differential equations.
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Advances in Difference Equations
In the last years, a few authors have gradually paid more attentions to applying critical point
theory to deal with problems for nonlinear second discrete systems; we refer to 5–9. But all
these systems do not concern with the p-Laplacian. For the reader’s convenience, we recall the
definition of the weak closure.
w
w
Suppose that E ⊂ X. We denote E as the weak closure of E, that is, x ∈ E if there exists
a sequence {xn } ⊂ E such that Λxn →Λx for every Λ ∈ X ∗ .
Very recently, based on a new variational principle of Ricceri 10, the following three
critical points was established by Bonanno 11.
Theorem 1.1 see 11, Theorem 2.1. Let X be a separable and reflexive real Banach space.
Φ : X→R a nonnegative continuously Gâteaux differentiable and sequentially weakly lower
semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X ∗ . J : X→R
a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that
there exists x0 ∈ X such that Φx0 Jx0 0 and that
i lim x→∞ Φx − λJx ∞ for all λ ∈ 0, ∞;
Further, assume that there are r > 0, x1 ∈ X such that
ii r < Φx1 ;
w Jx < r/r Φx Jx .
iii sup
1
1
x∈ Φ−1 −∞,r
Then, for each
Φx1 r
λ ∈ Λ1 ,
1.2
,
w Jx sup
w Jx
Jx1 − sup
−1
−1
x∈Φ −∞,r
x∈Φ −∞,r
the equation
Φ x − λJ x 0
has at least three solutions in X and, moreover, for each h > 1, there exists an open interval
hr
Λ2 ⊆ 0,
w Jx
rJx1 /Φx1 − sup
−1
1.3
1.4
x∈Φ −∞,r
and a positive real number σ such that, for each λ ∈ Λ2 , 1.3 has at least three solutions in X whose
norms are less than σ.
Here, our principle aim is by employing Theorem 1.1 to establish the existence of at least
three solutions for the p-Laplacian discrete boundary value problem 1.1.
The paper is organized as follows. The next section is devoted to give some basic definitions. In Section 3, under suitable hypotheses, we prove that the problem 1.1 possesses at
least three solutions when λ lies in exactly determined two open intervals, respectively; moreover, all these solutions are uniformly bounded with respect to λ belonging to one of the two
open intervals. At last, a consequence is presented.
2. Preliminaries
The class H of the functions x : Z0, T 1→R such that x0 xT 1 0 is a T -dimensional
Hilbert space with inner product
x, z T
k1
xkzk,
∀x, z ∈ H.
2.1
L. Jiang and Z. Zhou
3
We denote the induced norm by
x T
1/2
x k
2
x ∈ H.
,
2.2
k1
Furthermore, for any constant p > 1, we define other norms
xp xp T
1/p
|xk|
p
,
k1
T 1
∀x ∈ H,
2.3
1/p
|Δxk − 1|p
,
∀x ∈ H.
k1
Since H is a finite dimensional space, there exist constants c2p ≥c1p > 0 such that
c1p xp ≤ xP ≤ c2p xp .
2.4
The following two functionals will be used later:
Φx T 1
1
|Δxk − 1|p ,
p k1
Jx T
Fk, xk,
2.5
k1
ξ
where x ∈ H, Fk, ξ : 0 fk, sds for any ξ ∈ R. Obviously, Φ, J ∈ C1 H, R, that is, Φ and
J are continuously Fréchet differentiable in H. Using the summation by parts formula and the
fact that x0 xT 1 0 for any x ∈ H, we get
Φ xz lim
t→0
T 1
T 1
Φx tz − Φx |Δxk − 1|p−2 Δxk − 1Δzk − 1
t
k1
φp Δxk − 1Δzk − 1
k1
T
φp Δxk − 1Δzk − 1 − φp ΔxT zT k1
φp Δxk − 1zk − 1|T1 1 −
T
2.6
Δφp Δxk − 1zk − φp ΔxT zT k1
T
− Δφp Δxk − 1zk
k1
for any x, z ∈ H. Noticing the fact that x0 xT 1 0 for any x ∈ H again, we obtain
J xz lim
t→0
for any x, z ∈ H.
T
Jx tz − Jx fk, xkzk
t
k1
2.7
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Advances in Difference Equations
Remark 2.1. Obviously, for any x, z ∈ H,
Φ − λJ xz −
T
Δφp Δxk − 1 λfk, xk zk 0
2.8
k1
is equivalent to
Δφp Δxk − 1 λfk, xk 0
2.9
for any k ∈ Z1, T with x0 xT 1 0. That is, a critical point of the functional Φ − λJ
corresponds to a solution of the problem 1.1. Thus, we reduce the existence of a solution for
the problem 1.1 to the existence of a critical point of Φ − λJ on H.
The following estimate will play a key role in the proof of our main results.
Lemma 2.2. For any x ∈ H and p > 1, the relation
max {|xk|} ≤
k∈Z1,T T 1p−1/p
xP
2
2.10
holds.
Proof. Let τ ∈ Z1, T such that
|xτ| max {|xk|}.
k∈Z1,T 2.11
Since x0 xT 1 0 for any x ∈ H, by Cauchy-Schwarz inequality, we get
1/p
τ
τ
τ
p
1/q
|Δxk − 1| ≤ τ
|Δxk − 1|
,
|xτ| Δxk − 1
≤
k1
k1
k1
T 1
T 1
|xτ| Δxk − 1
≤
|Δxk − 1|
kτ1
kτ1
1/p
T 1
p
1/q
|Δxk − 1|
,
≤ T − τ 1
2.12
2.13
kτ1
for any x ∈ H, where q is the conjugative number of p, that is, 1/p 1/q 1.
If
τ
|Δxk − 1|p ≤
k1
T 1p−1
p
xP ,
2p τ p−1
2.14
jointly with the estimate 2.12, we get the required relation 2.10.
If, on the contrary,
τ
k1
|Δxk − 1|p >
T 1p−1
p
xP ,
2p τ p−1
2.15
L. Jiang and Z. Zhou
5
thus,
T 1
p
|Δxk − 1|p xP −
kτ1
τ
|Δxk − 1|p <
k1
T 1p−1
p
xP .
1−
p
p−1
2 τ
2.16
Combining the above inequality with the estimate 2.13, we have
T 1p−1 1/p
xP .
|xτ| < T − τ 11/q 1 −
2p τ p−1
2.17
Now, we claim that the inequality
T 1p−1 1/p T 1p−1/p
≤
T − τ 11/q 1 −
2
2p τ p−1
2.18
holds, which leads to the required inequality 2.10. In fact, we define a continuous function
υ :0, T 1→R by
υs 1
T − s 1
p−1
1
.
sp−1
2.19
This function υ can attain its minimum 2p /T 1p−1 at s T 1/2. Since τ ∈ Z1, T , we
have υτ≥2p /T 1p−1 , namely,
2p
T 1p−1
≤
1
T − τ 1p−1
1
.
τ p−1
2.20
This implies the assertion 2.18. Lemma 2.2 is proved.
3. Main results
First, we present our main results as follows.
ξ
Theorem 3.1. Let fk, · ∈ CR, R for any k ∈ Z1, T . Put Fk, ξ 0 fk, sds for any ξ ∈ R
and assume that there exist four positive constants c, d, μ, α with c < T 1/2p−1/p d and α < p
such that
A1 max k,ξ∈Z1,T ×−c,c Fk, ξ < 2cp /T 2cp 2T 1p−1 dp Tk1 Fk, d;
A2 Fk, ξ ≤ μ1 |ξ|α .
Furthermore, put
ϕ1 ϕ2 pT 1p−1 T max k,ξ∈Z1,T ×−c,c Fk, ξ
,
2cp
T
p
Fk,
d
−
T
max
Fk,
ξ
k,ξ∈
Z
1,T
×−c,c
k1
2dp
3.1
,
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Advances in Difference Equations
and for each h > 1,
a
2p−1 pc
h2cdp
T
p
k1 Fk,
d − T T 1p−1 pdp max k,ξ∈Z1,T ×−c,c Fk, ξ
3.2
.
Then, for each
1 1
,
λ ∈ Λ1 ,
ϕ2 ϕ1
3.3
the problem 1.1 admits at least three solutions in H and, moreover, for each h > 1, there exist an open
interval Λ2 ⊆ 0, a and a positive real number σ such that, for each λ ∈ Λ2 , the problem 1.1 admits
at least three solutions in H whose norms in H are less than σ .
Remark 3.2. By the condition A1 , we have
T 2cp 2T 1p−1 dp Fk, ξ < 2cp
max
k,ξ∈Z1,T ×−c, c
T
3.4
Fk, d.
k1
That is,
2d T 1
p
p−1
T
max
Fk, ξ < 2c
p
T
k,ξ∈Z1,T ×−c,c
Fk, d − T
k1
max
3.5
Fk, ξ .
k,ξ∈Z1,T ×−c,c
Thus, we get
p−1
pT 1
T max k, ξ∈Z1, T ×−c, cFk, ξ
<
2cp
p
T
k1 Fk,
d − T max k, ξ∈Z1, T ×−c, cFk, ξ
2dp
3.6
Namely, we obtain the fact that ϕ1 < ϕ2 .
Proof of Theorem 3.1. Let X be the Hilbert space H. Thanks to Remark 2.1, we can apply
Theorem 1.1 to the two functionals Φ and J. We know from the definitions in 2.5 that Φ
is a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X ∗ , and J is a
continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Now,
put x0 k 0 for any k ∈ Z0, T 1, it is easy to see that x0 ∈ H and Φx0 Jx0 0.
Next, in view of the assumption A2 and the relation 2.4, we know that for any x ∈ H
and λ≥0,
Φx − λJx T 1
T
1
|Δxk − 1|p − λ Fk, xk
p k1
k1
T
1
p
≥ xP − λμ 1 |xk|α p
k1
p
T
c1p
p
α
≥
|xk| − λμ |xk| − λμ .
p
k1
3.7
L. Jiang and Z. Zhou
7
Taking into account the fact that α < p, we obtain, for all λ ∈ 0, ∞,
lim Φx − λJx ∞.
3.8
x→∞
The condition i of Theorem 1.1 is satisfied.
Now, we let
⎧
⎪
⎪
⎨0,
x1 k d,
⎪
⎪
⎩0,
k 0,
k ∈ Z1, T ,
2c
r
3.9
k T 1.
p
pT 1p−1
.
It is clear that x1 ∈ H,
Φx1 Jx1 T 1
1
2dp
,
|Δxk − 1|p p k1
p
T
Fk, x1 k k1
T
3.10
Fk, d.
k1
In view of c < T 1/2p−1/p d, we get
Φx1 2dp
2cp
r.
>
p
pT 1p−1
3.11
So, the assumption ii of Theorem 1.1 is obtained. Next, we verify that the assumption
iii of Theorem 1.1 holds. From Lemma 2.2, the estimate Φx ≤ r implies that
|xk|p ≤
pT 1p−1
prT 1p−1
T 1p−1
p
x
Φx
≤
P
2p
2p
2p
3.12
for any k ∈ Z1, T . From the definition of r, it follows that
Φ−1 − ∞, r ⊆ {x ∈ H : |xk| ≤ c, ∀k ∈ Z1, T }.
3.13
Thus, for any x ∈ H, we have
sup
w
x∈Φ−1 −∞,r
Jx sup
Jx ≤ T
x∈Φ−1 −∞,r
max
Fk, ξ.
k,ξ∈Z1,T ×−c,c
3.14
On the other hand, we get
T
r
2cp
Fk, d.
Jx1 r Φx1 2cp 2T 1p−1 dp k1
3.15
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Advances in Difference Equations
Therefore, it follows from the assumption A1 that
sup
w
Jx ≤
x∈Φ−1 −∞,r
r
Jx1 ,
r Φx1 3.16
that is, the condition iii of Theorem 1.1 is satisfied.
Note that
Φx1 Jx1 − sup
−1
w
x∈Φ −∞,r
Jx
1
2dp
,
≤ T
ϕ
2
p
Fk,
d
−
T
max
Fk,
ξ
k,ξ∈
Z
1,T
×−c,c
k1
r
sup
w
x∈Φ−1 −∞,r
Jx
≥
2cp
pT 1
p−1
T maxk,ξ∈Z1,T ×−c,c Fk, ξ
3.17
1
.
ϕ1
By a simple computation, it follows from the condition A1 that ϕ2 > ϕ1 . Applying
Theorem 1.1, for each λ ∈ Λ1 1/ϕ2 , 1/ϕ1 , the problem 1.1 admits at least three solutions in
H.
For each h > 1, we easily see that
hr
rJx1 /Φx1 − sup
≤
2p−1 pcp
T
w
x∈Φ−1 −∞,r
Jx
3.18
h2cdp
k1 Fk, d
− T T 1p−1 pdp max k,ξ∈Z1,T ×−c,c Fk, ξ
a.
Taking the condition A1 into account, it forces that a > 0. Then from Theorem 1.1, for each
h > 1, there exist an open interval Λ2 ⊆ 0, a and a positive real number σ, such that, for
λ ∈ Λ2 , the problem 1.1 admits at least three solutions in H whose norms in H are less than
σ. The proof of Theorem 3.1is complete.
As a special case of the problem 1.1, we consider the following systems:
Δ φp Δxk − 1 λwkgxk 0,
k ∈ Z1, T ,
x0 0 xT 1,
3.19
where w : Z1, T →R and g ∈ CR, R are nonnegative. Define
Wk k
wt,
Gξ t1
Then Theorem 3.1 takes the following simple form.
ξ
gsds.
0
3.20
L. Jiang and Z. Zhou
9
Corollary 3.3. Let w : Z1, T →R and g ∈ CR, R be two nonnegative functions. Assume that there
exist four positive constants c, d, η, α with c < T 1/2p−1/p d and α < p such that
A1 max k∈Z1, T wk < 2cp WT /T 2cp 2T 1p−1 dp Gd/Gc;
A2 Gξ≤η1 |ξ|α for any ξ ∈ R.
Furthermore, put
ϕ1 ϕ2 pT 1p−1 T Gc max k∈Z1,T wk
,
2cp
3.21
pWT Gd − T Gc max k∈Z1,T wk
,
2dp
and for each h > 1,
a
2cdp h
2p−1 pcp WT Gd − pdp T T 1p−1 Gc max k∈Z1,T wk
.
3.22
Then, for each
λ ∈ Λ1 1 1
,
,
ϕ2 ϕ1
3.23
the problem 3.19 admits at least three solutions in H and, moreover, for each h > 1, there exist an
open interval Λ2 ⊆ 0, a and a positive real number σ such that, for each λ ∈ Λ2 , the problem 3.19
admits at least three solutions in H whose norms in H are less than σ.
Proof. Note that from fact fk, s wkgs for any k ∈ Z1, T × R, we have
Fk, ξ Gc max wk.
max
k,ξ∈Z1,T ×−c, c
k∈Z1,T 3.24
On the other hand, we take μ η max k∈Z1,T wk. Obviously, all assumptions of Theorem 3.1
are satisfied.
To the end of this paper, we give an example to illustrate our main results.
Example 3.4. We consider 1.1 with fk, s kgs, T 15, p 3, where
gs s ≤ 4d,
es ,
se
4d
− 4d,
s > 4d.
We have that Wk 1/2kk 1 and
⎧
⎨eξ − 1,
ξ ≤ 4d,
Gξ 1 2
⎩ ξ e4d − 4dξ 1 − 4de4d 8d2 − 1, ξ > 4d.
2
3.25
3.26
It can be easily shown that, when c 1, d 15, η e60 , and α 2, all conditions of
Corollary 3.3 are satisfied.
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Advances in Difference Equations
Acknowledgments
This work is supported by the National Natural Science Foundation of China no. 10571032
and Doctor Scientific Research Fund of Jishou university no. jsdxskyzz200704.
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