Solutions of 2 th-Order Boundary Value Problem for Difference Equation via Variational Method

Difference equations have been applied as models in vast areas such as finance insurance, biological populations, disease control, genetic study, physical field, and computer application technology. Because of their importance, many literature deals with its existence and uniqueness problems. For example, see [1–10].

We notice that the existing results are usually obtained by various analytical techniques, for example, the conical shell fixed point theorem [1, 6], Banach contraction map method [7], Leray-Schauder fixed point theorem [2, 10], and the upper and lower solution method [3]. It seems that the variational technique combining with the critical point theory [11] developed in the recent decades is one of the effective ways to study the boundary value problems of difference equations. However because the variational method requires a "symmetrical'' functional, it is hard for the odd-order difference equations to create a functional satisfying the "symmetrical'' property. Therefore, the even-order difference equations have been investigated in most references.

Let , be integers, and , be a discrete interval in . Inspired by [5, 8], in this paper, we try to investigate the following th-order boundary value problem (BVP) of difference equation via variational method combining with some traditional analytical skills:

(11)

(12)

where is the forward difference operator; for and A variational functional for BVP (1.1)-(1.2) is constructed which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points of this functional. In order to prove the existence criteria of critical points of the functional, some lemmas are given in Section 2. Two criteria for the existence of at least one solution and two solutions for BVP (1.1)-(1.2) are established in Section 3 which is the generalization for BVP of the even-order difference equations. The existence results obtained in this paper are not found in the references, to the best of our knowledge.

For convenience, we will use the following notations in the following sections:

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We need two lemmas from [12] or [11].

Lemma 2.1.

Let be a real reflexive Banach space with a norm , and let be a weakly lower (upper) semicontinuous functional, such that

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then there exists such that

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Furthermore, if has bounded linear Gâteaux derivative, then

Lemma 2.2 (mountain-pass lemma).

Let be a real Banach space, and let be continuously differential, satisfying the P-S condition. Assume that and is an open neighborhood of , but If then is the critical value of where

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This means that there exists , s.t. ,

The following lemma will be used in the proof of Lemma 2.4.

Lemma 2.3.

If is a symmetric and positive-defined real matrix, is a real matrix, is the transposed matrix of Then is positive defined if and only if

Proof.

Since is positive defined, then

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Let be a Hilbert space defined by

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with the norm

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Hence is an -dimensional Hilbert space. For any , let then one can show that there exist constants s.t. ; that is, is an equivalent norm of (see [9, page 68]).

Lemma 2.4.

There is

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Proof.

Since , , By using the inequality , , we have

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Repeating the above process, we obtain

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On the other hand, define , where is the combination number, then we can rewrite in a vector form, that is, where , , and

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where Hence Note that

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and by Lemma 2.3 with , we know that is positive defined. Hence all eigenvalues of are real and positive. Let be the minimal eigenvalue of these eigenvalues, then Therefore that is,

However, how to find the in Lemma 2.4 is a skillful and challenging task. The following lemma from [13] offers some help for the estimation of .

Lemma 2.5 (Brualdi [13]).

If is weak irreducible, then each eigenvalue is contained in the set

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In Lemma 2.5, is the complex number set, and the denotations , , , can be found in [13]. Since is positive defined, all eigenvalues are positive real numbers. Therefore, by Lemma 2.5, let

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where . is a subset of and can be calculated directly from Define If , we can use this as in Lemma 2.4. If , then one needs to calculate the eigenvalues directly.

Define the functional on by

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Then is with

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where and is the inner product in . In fact, we have

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The continuity of and the right-hand side of the inequality in Lemma 2.4 lead to (2.15).

Furthermore, for any we have , By using the following formula (e.g., see [14, page 28]):

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we have

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Repeating the above process, we obtain

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Let , that is,

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Since is arbitrary, we know that the solution of BVP (1.1)-(1.2) corresponds to the critical point of

Now we present our main results of this paper.

Theorem 3.1.

If there exist , , , and s.t.

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then BVP (1.1)-(1.2) has at least one solution.

Proof.

In fact, we can choose a suitable such that

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Since there exists with we have

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Then by Lemma 2.4, we obtain

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Noticing that we have From Lemma 2.1, the conclusion of this lemma follows.

Corollary 3.2.

If there exists s.t. for all , and

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where , satisfy either , or , then BVP (1.1)-(1.2) has at least one solution.

Proof.

Assume that , Then for there exists such that as . We have from the continuity of that there is a such that for all , . When , one has for then

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when , one has for then

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Let , then we have for Therefore, we have

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which implies and by Lemma 2.1, the conclusion of this lemma follows.

Assume that , Then for there exists such that as . We have from the continuity of that there is a such that for all , . When , one has , then we have

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when , one has , then we have

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Let , then we have for Therefore, by Theorem 3.1, the conclusion of this lemma follows.

Theorem 3.3.

Assume that , and

1. (i)

   , is defined in Lemma 2.4;

    
2. (ii)

   satisfies (3.1) in Theorem 3.1 or satisfies the assumptions in Corollary 3.2.

    

Then BVP (1.1)-(1.2) has at least two solutions.

Proof.

We first show that satisfies the P-S condition. Let satisfy that is bounded and If is unbounded, it possesses a divergent subseries, say as . However from (ii), we get (3.4) or (3.8), hence as , which is contradictory to the the fact that is bounded.

Next we use the mountain-pass lemma to finish the proof. By (i), for , there exists such that for . Then for . Now together with Lemma 2.3 we have

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which implies that

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where is the zero element in , and Since we have from (3.4) or (3.8) that there exists with that is, but Using Lemma 2.2, we have shown that is the critical value of with defined as

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We denote as its corresponding critical point.

On the other hand, by Theorem 3.1 or Corollary 3.2, we know that there exists s.t. If the theorem is proved. If on the contrary, , that is, that implies for any , Taking in with , by the continuity of there exist s.t. , Hence , are two different critical points of that is, BVP (1.1)-(1.2) has at least two different solutions.

Consider the 6th-order boundary value problem for difference equation

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Let , we have , Hence satisfies the conditions in Theorem 3.3, the boundary value problem (4.1) has at least two solutions.