Three Solutions to Dirichlet Boundary Value Problems for -Laplacian Difference Equations

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:

(1.1)

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

(1.3)

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

(1.4)

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

(2.1)

We denote the induced norm by

(2.2)

Furthermore, for any constant , we define other norms

(2.3)

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

(2.4)

The following two functionals will be used later:

(2.5)

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

(2.6)

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

(2.7)

for any .

Remark 2.1.

Obviously, for any ,

(2.8)

is equivalent to

(2.9)

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

(2.10)

holds.

Proof.

Let such that

(2.11)

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

(2.12)

(2.13)

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

If

(2.14)

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

If, on the contrary,

(2.15)

thus,

(2.16)

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

(2.17)

Now, we claim that the inequality

(2.18)

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

(2.19)

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

(2.20)

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

(3.1)

and for each ,

(3.2)

Then, for each

(3.3)

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

(3.4)

That is,

(3.5)

Thus, we get

(3..6)

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 ,

(3.7)

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

(3.8)

The condition (i) of Theorem 1.1 is satisfied.

Now, we let

(3.9)

It is clear that ,

(3.10)

In view of , we get

(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 implies that

(3.12)

for any . From the definition of , it follows that

(3.13)

Thus, for any , we have

(3.14)

On the other hand, we get

(3.15)

Therefore, it follows from the assumption (A₁) that

(3.16)

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

Note that

(3.17)

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

(3.18)

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:

(3.19)

where and are nonnegative. Define

(3.20)

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

(3.21)

and for each ,

(3.22)

Then, for each

(3.23)

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

(3.24)

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

(3.25)

We have that and

(3.26)

It can be easily shown that, when , and , all conditions of Corollary 3.3 are satisfied.