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Three Solutions to Dirichlet Boundary Value Problems for
-Laplacian Difference Equations
Advances in Difference Equations volume 2008, Article number: 345916 (2007)
Abstract
We deal with Dirichlet boundary value problems for -Laplacian difference equations depending on a parameter
. Under some assumptions, we verify the existence of at least three solutions when
lies in two exactly determined open intervals respectively. Moreover, the norms of these solutions are uniformly bounded in respect to
belonging to one of the two open intervals.
1. Introduction
Let be all real numbers, integers, and positive integers, respectively. Denote
and
with
for any
.
In this paper, we consider the following discrete Dirichlet boundary value problems:

where is a positive integer,
is a constant,
is the forward difference operator defined by
,
is a
-Laplacian operator, that is,
,
for any
.
There seems to be increasing interest in the existence of solutions to boundary value problems for finite difference equations with -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. 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
-Laplacian. For the reader's convenience, we recall the definition of the weak closure.
Suppose that . We denote
as the weak closure of
, that is,
if there exists a sequence
such that
for every
.
Very recently, based on a new variational principle of Ricceri [10], the following three critical points was established by Bonanno [11].
Theorem(see 1.1 (see {[11, Theorem 2.1]}).
Let be a separable and reflexive real Banach space.
a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on
.
a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there exists
such that
and that
-
(i)
for all
;
Further, assume that there are
such that
-
(ii)
;
-
(iii)
.
Then, for each
(1.2)
the equation

has at least three solutions in and, moreover, for each
, there exists an open interval

and a positive real number such that, for each
, (1.3) has at least three solutions in
whose norms are less than
.
Here, our principle aim is by employing Theorem 1.1 to establish the existence of at least three solutions for the -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 lies in exactly determined two open intervals, respectively; moreover, all these solutions are uniformly bounded with respect to
belonging to one of the two open intervals. At last, a consequence is presented.
2. Preliminaries
The class of the functions
such that
is a
-dimensional Hilbert space with inner product

We denote the induced norm by

Furthermore, for any constant , we define other norms

Since is a finite dimensional space, there exist constants
such that

The following two functionals will be used later:

where ,
for any
. Obviously,
, that is,
and
are continuously Fréchet differentiable in
. Using the summation by parts formula and the fact that
for any
, we get

for any . Noticing the fact that
for any
again, we obtain

for any .
Remark 2.1.
Obviously, for any ,

is equivalent to

for any with
. That is, a critical point of the functional
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
on
.
The following estimate will play a key role in the proof of our main results.
Lemma 2.2.
For any and
, the relation

holds.
Proof.
Let such that

Since for any
, by Cauchy-Schwarz inequality, we get


for any , where
is the conjugative number of
, that is,
.
If

jointly with the estimate (2.12), we get the required relation (2.10).
If, on the contrary,

thus,

Combining the above inequality with the estimate (2.13), we have

Now, we claim that the inequality

holds, which leads to the required inequality (2.10). In fact, we define a continuous function by

This function can attain its minimum
at
. Since
, we have
, namely,

This implies the assertion (2.18). Lemma 2.2 is proved.
3. Main Results
First, we present our main results as follows.
Theorem 3.1.
Let for any
. Put
for any
and assume that there exist four positive constants
with
and
such that
(A1)
(A2) .
Furthermore, put

and for each ,

Then, for each

the problem (1.1) admits at least three solutions in and, moreover, for each
, there exist an open interval
and a positive real number
such that, for each
, the problem (1.1) admits at least three solutions in
whose norms in
are less than
.
Remark 3.2.
By the condition (A1), we have

That is,

Thus, we get

Namely, we obtain the fact that .
Proof of Theorem 3.1.
Let be the Hilbert space
. Thanks to Remark 2.1, we can apply Theorem 1.1 to the two functionals
and
. We know from the definitions in (2.5) that
is a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on
, and
is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Now, put
for any
, it is easy to see that
and
.
Next, in view of the assumption (A2) and the relation (2.4), we know that for any and
,

Taking into account the fact that , we obtain, for all
,

The condition (i) of Theorem 1.1 is satisfied.
Now, we let

It is clear that ,

In view of , we get

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 implies that

for any . From the definition of
, it follows that

Thus, for any , we have

On the other hand, we get

Therefore, it follows from the assumption (A1) that

that is, the condition (iii) of Theorem 1.1 is satisfied.
Note that

By a simple computation, it follows from the condition (A1) that . Applying Theorem 1.1, for each
, the problem (1.1) admits at least three solutions in
.
For each , we easily see that

Taking the condition (A1) into account, it forces that . Then from Theorem 1.1, for each
, there exist an open interval
and a positive real number
, such that, for
, the problem (1.1) admits at least three solutions in
whose norms in
are less than
. The proof of Theorem 3.1is complete.
As a special case of the problem (1.1), we consider the following systems:

where and
are nonnegative. Define

Then Theorem 3.1 takes the following simple form.
Corollary 3.3.
Let and
be two nonnegative functions. Assume that there exist four positive constants
with
and
such that
(A′1)
(A′2) for any
.
Furthermore, put

and for each ,

Then, for each

the problem (3.19) admits at least three solutions in and, moreover, for each
, there exist an open interval
and a positive real number
such that, for each
, the problem (3.19) admits at least three solutions in
whose norms in
are less than
.
Proof.
Note that from fact for any
, we have

On the other hand, we take . 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 , where

We have that and

It can be easily shown that, when , and
, all conditions of Corollary 3.3 are satisfied.
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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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Jiang, L., Zhou, Z. Three Solutions to Dirichlet Boundary Value Problems for -Laplacian Difference Equations.
Adv Differ Equ 2008, 345916 (2007). https://doi.org/10.1155/2008/345916
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DOI: https://doi.org/10.1155/2008/345916
Keywords
- Differential Equation
- Hilbert Space
- Banach Space
- Continuous Function
- Partial Differential Equation