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Discrete Dynamics in Nature and Society
VolumeΒ 2011, Article IDΒ 319829, 15 pages
http://dx.doi.org/10.1155/2011/319829
Research Article

Existence and Multiplicity of Solutions for Discrete Nonlinear Two-Point Boundary Value Problems

School of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037008, China

Received 26 September 2010; Accepted 5 January 2011

Academic Editor: Chang-hongΒ Wang

Copyright Β© 2011 Jianmin Guo and Caixia Guo. 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

By using Morse theory, the critical point theory, and the character of , we consider the existence and multiplicity results of solutions to the following discrete nonlinear two-point boundary value problem subject to , where is a positive integer, is the forward difference operator defined by , and is continuous. In argument, Morse inequalities play an important role.

1. Introduction

In this paper, we discuss the existence and multiplicity results of solutions to the following discrete nonlinear two-point boundary value problem (BVP): where is a positive integer, is the forward difference operator defined by , and is continuous.

Recent years, there have been many papers studying the existence and multiplicity of solutions for differential equations. For example, by employing the strongly monotone operator principle and the critical point theory, F. Li et al. in [1] establish some conditions on which are able to guarantee a class of boundary value problem on differential equation has a unique solution, at least one nonzero solution and infinitely many solutions; by using the critical point theory, and Morse theory, Yang and Zhang in [2] obtained some existence results for differential equations with parameters; there are also many authors who studied the existence results of positive solutions to boundary value problem on differential equation by employing the cone expansion or compression fixed point theorem or the critical point theory, see [3–5]; Jiang and Zhou in [6] obtained variational framework of the BVP(1.1) first by virtue of Green's function and separation of linear operator and studied the existence of a unique solution or at least one nontrivial solution by employing the strongly monotone operator principle and the critical point theory respectively.

In this paper, the main difference from the ordinary literatures is that we apply Morse theory and critical point theory to deal with problems on discrete systems. Then we establish some conditions on which include sublinear, superlinear or asymptotical case to guarantee that BVP(1.1) has at least one solution, at least two nontrivial solutions, and infinitely many solutions.

2. Preliminary

In this section, we give some notations and lemmas.

Let is a -dimensional Hilbert space with inner product , and we denote the induced norm by . . And for any and . Let be a real Banach space, and let denote the set of functionals that are FrΓ©chet differentiable and their FrΓ©chet derivatives are continuous on .

Let [6] be the Green's function of the linear boundary value problem where then the (BVP)(1.1) has a solution if and only if the following equation: has a solution in . Define operators by It is easy to see that a solution of (2.3) is equivalent to a solution in of the following operator equation:

It is well know, that all eigenvalues of are which have the corresponding orthonormal eigenfunctions

Remark 2.1 (see [6]). (i) is a linear continuous operator; furthermore, is symmetric in .(ii)There are .(iii)The square root operator of : is bounded linear and symmetric, (iv), , this implies that for all with . Therefore, for all with .

In next section, we will use the critical point theory and Morse theory to discuss the main results. Here, we state some necessary definitions and lemmas.

Definition 2.2. Let be a real Banach space; is an open subset of . Suppose the functional is on . If and the FrΓ©chet derivative , then we call that is a critical point of the functional , and is a critical value of .

Definition 2.3. Let . If any sequence for which is bounded and as possesses a convergent subsequence, then we say that satisfies Palais-Smale condition (denoted by P.S. condition for short).

Lemma 2.4 (see [7, Definition 1.16, page 13]). Let be a real reflexive Banach space. Suppose that the functional is , and is bounded below and satisfies P.S. condition, then must have a minimum in , that is, there exists a such that , therefore is a critical point of the functional .

Definition 2.5 (see [2]). Let , with be a real Banach space. Assume that there exists small, such that Then has a local linking at 0.

Definition 2.6. Let be a real Banach space, let be an isolated critical point of with and let be a neighborhood of , containing the unique critical point. We call the -th critical group of at , where stands for the -th singular relative homology group with integer coefficients. We say that is a homological nontrivial critical point of if at least one of its critical groups is nontrivial.

Lemma 2.7 (see [8]). Assume that satisfies P.S. condition and has a local linking at 0. Then , that is, 0 is a homological nontrivial critical point of .

Lemma 2.8 (see [8]). Let be a Finsler manifold of . Assume that satisfies the P.S. condition and is the only critical value of in . If connected component of is only composed of isolated critical points, then is a deformation retract of .

Lemma 2.9 (see [2]). (i) The operator equation has a solution in if and only if the operator equation has a solution in .
(ii) The uniqueness of solution for these two above equations is also equivalent.
(iii) If (2.11) has a nonzero solution in , then (2.10) has a nonzero solution in . If (2.11) has infinitely many solutions in , then (2.10) has also infinitely many solutions in .

Lemma 2.10 (see [9]). Suppose that the functional has a critical point , where , then (BVP)(1.1) has a solution in .

Lemma 2.11 (see [8]). Suppose satisfies P.S. and (A) conditions, then, one has where . Furthermore, if the left of the equation is convergent, then one has that
(A) Suppose that there are two regular values, then has at most finite critical points and rank of critical groups of every critical point is finite.

Lemma 2.12 (see [8, page 100, Theorem 3.2]). Let be a functional and satisfy P.S. condition. Suppose that , where is a compact map and is a isolated critical point of , then we have

3. Mail Results

Lemma 3.1. Suppose satisfies one of the following conditions:(a)for , where ,(b) for , where ; and ( for , where ,(c) and (, uniformly for , where , then one has(i) is coercive on , that is as , (ii) satisfies P.S. condition.

Proof. (i)Let (a) hold. It follows from that there is a constant such that . Therefore,

Let (b) or (c) hold. By the condition (, write , . If holds, then ; if holds, then , . It follows that for every , there is such that Integrating the equality over the interval , we have Letting , we see that , , . In a similar way, we have , , . Hence, .

Let be such that as and for some constant . Taking , then , and . So, in . Then, Hence and . And , as for . Therefore, This is impossible, so is coercive on .(ii)Let is bounded and . By (i), is bounded on . Clearly, possesses a convergent subsequence. Then, the P.S. condition is satisfied.

Lemma 3.2. Suppose that ( there exist and an integer , which satisfy such that .
Then the functional has a local linking with respect to , where .

Proof. Let . If , then ; if , then .
By Remark 2.1, we can assume that for the given , there exists a such that , , and thus, by (, we have For , consider the above . By Remark 2.1, we still have ; thus by (, we have This implies that has a local linking at 0 with respect to .

Theorem 3.3. If condition ( holds, then the BVP(1.1) has at least a solution.

Proof. We will verify that the functional defined in Lemma 2.10 has a critical point . By [6], we know that is functional. And by Lemma 3.1, it is easy to know that is bounded below and satisfies P.S. condition. It follows from Lemma 2.4 that has a critical point in .

Theorem 3.4. If conditions ( and ( hold, then the BVP(1.1) has at least a solution.

Theorem 3.5. If conditions ( and ( hold, then the BVP(1.1) has at least a solution. The proof of Theorem 3.4 and Theorem 3.5 is similar to the proof of Theorem 3.3.

Theorem 3.6. If conditions ( and ( hold, then the BVP(1.1) has at least two nontrivial solutions.

Proof. By Lemma 3.1, we know that satisfies P.S. condition, and it follows from Lemmas 2.7 and 3.2 that is a homological nontrivial critical point of , then . If , then , , , which implies that all with are solutions of the BVP(1.1). If , that is, 0 is not a minimizer of , by Lemma 3.1, it is easy to know that is bounded below and satisfies P.S. condition. Then by Lemma 2.4, has a minimizer , that is, a critical point and ; Without loss of generality, we may suppose that the minimizer is unique. Let , then Morse index of is
We suppose, only have two critical points with and , where , , . Let , is small enough such that .
If is finite, then satisfies additivity. So, It follows from the second deformation lemma, exactness, and excision that . Thus, Hence, .
By the same way, So, . But Then, is a contradict with (3.12).
Thus, has at least three critical points, that is, the BVP (1.1) has at least two nontrivial solutions.

Theorem 3.7. If condition (, (, and ( hold, then the BVP(1.1) has at least two nontrivial solutions.

Theorem 3.8. If condition (, (, and ( hold, then the BVP(1.1) has at least two nontrivial solutions.

The proof of Theorems 3.5, and 3.6 is similar to the proof of Theorem 3.6.

Theorem 3.9. Suppose that (H6) there exists , such that (H7) there exist and such that for all and with . Then the BVP(1.1) has at least one nontrivial solution.

Proof. Obviously, and , where . It follows from (H6) that there exists a sufficiently small such that Since for all , for . So, for every , where is -dimension subspace of . Therefore, , where -th eigenvalues of are . If there exists such that , then . Thus, . It is not true. So, 1 is not eigenvalues of . Thus is a nondegenerate critical point of and Morse index number is . Then, .
Asumme that is no other critical points, then Morse index number of is
It follows from ( that there exists such that , . So, This implies for all . So, there exists such that for all . Then, we will prove that there exists such that , and if , then .
Indeed, let , then for all , So, satisfies and for all is unique. It follows from implicit function theorem that we have , where .
At last, and there exists such that and for all . We define a deformation retract by This proves that . The,n . Therefore, . Since is contractible, we have Betti number . Thus, Then by (3.19) and (3.23), we have . Thus, according to the Morse inequality, has at least one nontrivial critical point.

Theorem 3.10. Suppose that condition ( is satisfied and that ( is odd in , that is, for all ,. Then the BVP(1.1) has infinitely many solutions.

Proof. First, it follows from ( that satisfies P.S. condition in [6].
It follows from ( that is even. Asumme that has only finitely many critical points . We choose two constant numbers and , such that they satisfy and .
On the one hand, it follows from Theorem 3.9 that . Then, . Since is contractible, we can deduce that Betti number .
Therefore, On the other hand, we choose enough small such that , , , is mutually disjoint. So, It follows from Borsuk theorem that that is By (3.24) and (3.27), we obtain that It is a contradiction. Thus, has infinitely many critical points in .

Example 3.11. Consider the (BVP) Here, . By simple calculation, we have . Thus, the condition ( holds. Then by Theroem 3.3., the (BVP) has at least one solution.

Example 3.12. Consider the (BVP) of Example 3.11, here

By simple calculation, we have and there exists and such that Let , then satisfy Thus, the conditions ( and ( hold. Hence, by Theorem 3.6, the (BVP) has at least two nontrivial solutions.

Example 3.13. Consider the (BVP) of Example 3.11, where

It is obvious that is odd with respect to . On the other hand, and we have It is easy to prove that condition ( holds. Hence, by Theorem 3.10, the (BVP) has infinitely many solutions.

Acknowledgment

The authors greatly thank Professor Fuyi Li for his help and many valuable discussions.

References

  1. F. Li, Y. Li, and Z. Liang, β€œExistence of solutions to nonlinear Hammerstein integral equations and applications,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 209–227, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. Y. Yang and J. Zhang, β€œExistence of solutions for some fourth-order boundary value problems with parameters,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1364–1375, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. G. Han and Z. Xu, β€œMultiple solutions of some nonlinear fourth-order beam equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 12, pp. 3646–3656, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. Y. Li, β€œPositive solutions of fourth-order boundary value problems with two parameters,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 477–484, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  5. Z. Bai and H. Wang, β€œOn positive solutions of some nonlinear fourth-order beam equations,” Journal of Mathematical Analysis and Applications, vol. 270, no. 2, pp. 357–368, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  6. L. Jiang and Z. Zhou, β€œExistence of nontrivial solutions for discrete nonlinear two point boundary value problems,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 318–329, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  7. D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Ji'nan, China, 2001.
  8. K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, vol. 6 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1993.
  9. F. Li, Y. Li, and Z. Liang, β€œExistence and multiplicity of solutions to 2mth-order ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 958–977, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet