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Discrete Dynamics in Nature and Society
Volume 2009, Article ID 407623, 14 pages
http://dx.doi.org/10.1155/2009/407623
Research Article

Existence of Solutions to Boundary Value Problems for the Discrete Generalized Emden-Fowler Equation

Department of Computation Science, Zhongkai University of Agriculture and Engineering, Guangzhou, Guangdong 510225, China

Received 16 May 2009; Revised 30 October 2009; Accepted 29 November 2009

Academic Editor: Leonid Berezansky

Copyright © 2009 Tieshan He and Fengjian Yang. 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.

Abstract

We consider the existence of solutions to boundary value problems for the discrete generalized Emden-Fowler equation. By means of the minimax methods in the critical point theory, some new results are obtained. Two examples are also given to illustrate the main results.

1. Introduction and Statements of Main Results

Let and be the sets of all integers and real numbers, respectively. For , define ,   when .

In this paper, we consider the following boundary value problem (BVP for short) consisting of the discrete generalized Emden-Fowler equation: and the boundary value conditions: where is a positive integer, , , , and are constants, and is the forward difference operator. We assume that is nonzero and realvalued for each . We also assume that is realvalued for each ,   is realvalued for each , and is continuous in the second variable .

Equation (1.1) has been extensively studied by many authors; for example, see [19] concerning its disconjugacy, disfocality, oscillation, asymptotic behaviour, existence of periodic solutions, and solutions to boundary value problem.

Recently, Yu and Guo in [10] employed the critical point theory to obtain the existence of solutions to the BVP (1.1)-(1.2). Motivated by this and the results in [11], the main purpose of this paper is to give some new sufficient conditions for the existence of solutions to the BVP (1.1)-(1.2) by applying the Saddle Point Theorem and the Least Action Principle.

Before giving the main results, we first set The main results are as follows.

Theorem 1.1. Suppose that , satisfy the following assumptions.(F1) There are constants ,  ,   such that for all , (F2) One has(i)either   as , or(ii)  as ,where for all , . (P1) The matrix is singular.Then the BVP (1.1)-(1.2) has at least one solution.

Remark 1.2. There  are functions , , and satisfying our Theorem 1.1 and not satisfying the results in [10]. For example, let , be arbitrary constants, ,  ,   and where for every . It is easy to verify that (F1), (F2)(i), and (P1) are satisfied. Then the BVP (1.1)-(1.2) has at least one solution. And it is easy to see that this solution is a nonzero solution since .

Theorem 1.3. Suppose that , satisfy the following assumptions.(F3) For any ,  .(P2) The matrix is nonsingular.Then the BVP (1.1)-(1.2) has at least one solution.

Corollary 1.4. Suppose that , satisfy (F1) and (P2). Then the BVP (1.1)-(1.2) has at least one solution.

Remark 1.5. Since  for all ,   implies , our Corollary 1.4 extends Theorem in [10].

Theorem 1.6. Suppose that , satisfy the following assumptions.(F4) For any ,  .(F5) One has(i)either   as for all , or (ii)  as for all .(P3) The matrix is singular and indefinite. Then the BVP (1.1)-(1.2) has at least one solution.

Remark 1.7. There are functions ,  , and satisfying our Theorem 1.6 and not satisfying the results in [10]. For example, let , be arbitrary constants, , , and where for every . It is easy to verify that (F4), (F5)(i) and (P3) are satisfied. Then the BVP (1.1)-(1.2) has at least one solution. And it is easy to see that this solution is a nonzero solution since .

Corollary 1.8. Assume that (F4) holds. If one of the following conditions is satisfied: () the matrix is negative semi-definite, and (F5)(i) holds,  () the matrix is positive semi-definite, and (F5)(ii) holds, then, the BVP (1.1)-(1.2) has at least one solution.

2. Variational Structure and Two Basic Lemmas

Let be the real Euclidean space with dimension . For any , and denote the usual norm and inner product in , respectively.

Define the functional on as follows: It is well known that is a critical point of if and only if is a solution of the BVP (1.1)-(1.2), where and . For details, see [10]. It follows from the continuity of that is continuously differentiable on . Moreover, one has

When the matrix is singular and indefinite, we suppose that and are the positive and negative eigenvalues of , respectively, and . We also suppose that , and , are the eigenvectors of corresponding to eigenvalues , and , satisfying where , , ,  , , . We denote Then has the direct sum decomposition So, for each , can be expressed by where , is , or , respectively. Furthermore, we have the following estimates:

Set where and . Then for any ,

We will make use of the least action principle and saddle point theorem to obtain the critical points of . Let us first recall these theorems.

Lemma 2.1 (the least action principle, see [12]). Let be a real Banach space, and assume that is bounded from below in and satisfies the Palais-Smale condition ((PS) condition for short). Then is a critical value of .

Lemma 2.2 (saddle point theorem, see [13]). Let be a real Hilbert space, , where and is finite dimensional. Suppose that satisfies the (PS) condition and()there exist constants such that , where ,   denotes the boundary of ; ()there exist and a constant such that .Then possesses a critical value and where .

Remark 2.3. As shown in [14], a deformation lemma can be proved with the weaker condition () replacing the usual (PS) condition, and it turns out that the saddle point theorem holds under condition ().

3. Proofs of the Main Results

In order to prove Theorem 1.1, we need to prove the following lemma.

Lemma 3.1. Assume that conditions (F1), (F2), and (P3) hold. Then the functional (see (2.1)) satisfies the (PS) condition; that is, for any sequence such that is bounded and as , there exists a subsequence of which is convergent in .

Proof. First suppose that (F1), (F2)(i), and (P3) hold. Recall that is a finite dimensional Hilbert space. Consequently, in order to prove that satisfies the (PS) condition, we only need to prove that is bounded. Let be a sequence in such that is bounded and as . Then there exist and such that for all ,  .
Since is singular and indefinite, we write with where , respectively. By (F1), (2.9), and Hölder’ inequality (), we have where . On the other hand, by the fact that and are mutually orthogonal, one has . Hence we have Similarly to (3.3), we have
Take in (2.2). Then We know Thus, by (3.1) and (3.3), we have that is, It follows from (3.8) and that for all and some positive constants .
By (3.1), (2.9), (3.4), and (3.9), we have Since , we deduce for all and some positive constants . The above inequality and (F2)(i) imply that is bounded. Then it follows from (3.9) that is bounded. Thus we conclude that is bounded, and the (PS) condition is verified.
Now, suppose that (F1), (F2)(ii), and (P3) hold. By a similar argument as above, we know also that satisfies the (PS) condition. The proof is complete.

Proof of Theorem 1.1. Assume that (F1), (F2)(i), and (P1) hold. The proof for the case when (F1), (F2)(ii), and (P1) hold is similar and will be omitted here. Since is nonzero for each , the singular symmetric matrix has at least one nonzero eigenvalue and we will give the proof in three cases. (i)Suppose that the matrix is singular and indefinite. Then has the direct sum decomposition: In view of Lemma 3.1, we only to check that conditions and in the saddle point theorem hold. To this end, let , . For any , by (F1), (2.9), and the mean value theorem, we have where , . Since and (F2)(i), we have
On the other hand, for any , by (F1), (2.9), and the mean value theorem, we have where ,  . Since , we can obtain
Let , then it follows from (3.13) and (3.15) that and are satisfied. By the saddle point theorem, has at least one critical point.
(ii) Suppose that are the positive eigenvalues of and . Then has the direct sum decomposition: where and are defined as in (2.4) and (2.5), respectively. By a similar argument as in the proof of Lemma 3.1, we see that satisfies the (PS) condition. By (3.13), is bounded from below. Then, by the least action principle, is a critical value of .
(iii) Suppose that are the negative eigenvalues of and . Then has the direct sum decomposition: Following almost the same procedure as the proof of Lemma 3.1, we know also that satisfies the (PS) condition in this case.
For any , (3.15) holds. For any , since , we have Due to (F2)(i) and , By the saddle point theorem, there exists at least one critical point of .
Since has at least one critical point in all three cases, the BVP (1.1)-(1.2) has at least one solution.

Proof of Theorem 1.3. Since the matrix is nonsingular, we will give the proof in three cases.(i) Suppose that and are the positive and negative eigenvalues of , respectively, and . Then has the direct sum decomposition: Denote It follows from (F3) that there exists a positive constant such that for any .
We now prove that the functional satisfies the (PS) condition. Let be a sequence in such that is bounded and as . Write , where ,  . Similarly to (3.8), we have, by (3.18), Thus is bounded, and the (PS) condition is verified.
For any , by (3.18) and the mean value theorem, we have where ,  . For any , Similarly to (3.20), we have
Let , then it follows from (3.20) and (3.21) that and are satisfied. By the saddle point theorem, has at least one critical point.
(ii) Suppose that are the positive eigenvalues of and . Then has the direct sum decomposition: It follows from (3.20) that satisfies the (PS) condition and is bounded from below. Then, by the least action principle, is a critical value of .
(iii) Suppose that are the negative eigenvalues of and . Then has the direct sum decomposition: It follows from (3.21) that satisfies the (PS) condition and is bounded from above. Then, by the least action principle, is a critical value of .
Since has a critical point in all three cases, the BVP (1.1)-(1.2) has at least one solution.

Proof of Corollary 1.4. This is immediate from Theorem 1.3.

The following lemma is useful for proving Theorem 1.6 and Corollary 1.8.

Lemma 3.2. Under the condition (F5), the functional satisfies condition (); that is, for any sequence such that is bounded and as , there exists a subsequence of which is convergent in .

Proof. First suppose that (F5)(i) holds. Let be a sequence in such that is bounded and as . Then there exists a constant such that for all . Hence, we have Then, is bounded. In fact, if is unbounded, there exist a subsequence of (still denoted by ) and such that By (F5)(i), we have The continuity of with respect to and (F5)(i) implies that there exists a constant such that for any , Then, we get Thus, which contradicts (3.23). Therefore, is bounded in and satisfies condition ().
Now, suppose that (F5)(ii) holds. By a similar argument as above, we know also that satisfies condition (). The proof is complete.

Proof of Theorem 1.6. Assume that (F4), (F5)(i), and (P3) hold. The proof for the case when (F4), (F5)(ii), and (P3) hold is similar and will be omitted here. Due to (P3), has the direct sum decomposition: where and are defined as in (2.4) and (2.5), respectively. We claim that for every , Indeed, according to (F5)(i), we can obtain that for any given , there exists a positive constant such that for , Obviously, where ,  . We have By integrating both sides of the above inequality from to , we get Let in the above inequality, and it follows from (F4) that for . From the arbitrariness of , we can conclude that (3.27) holds, proving our claim.
Now we prove If (3.33) does not hold, there exist a constant and a sequence in such that as and for all . Since as , there exist a subsequence of (still denoted by ) and such that By (3.27), we have The continuity of with respect to and (3.27) implies that there exists a constant such that for any , Then, we get Thus, which contradicts (3.34). Hence, (3.33) follows.
On the other hand, by (F4), there exists a positive constant such that for any , Then we have for all . Thus, we can conclude that
It follows from (3.33) and (3.40) that satisfies conditions () and (). Hence Theorem 1.6 follows from Lemma 3.2, the saddle point theorem, and Remark 2.3.

The proof of Corollary 1.8 is similar to that of Theorem 1.6 and is omitted.

Acknowledgments

The authors thank the referees for their valuable comment and careful corrections. This project is supported by Science and Technology Plan Foundation of Guangzhou (No. 2006J1-C0341).

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