Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2010, Article ID 705387, 14 pages
http://dx.doi.org/10.1155/2010/705387
Research Article

Sign-Changing Solutions for Discrete Second-Order Three-Point Boundary Value Problems

1Department of Computation Science, Zhongkai University of Agriculture and Engineering, Guangzhou, Guangdong 510225, China
2Department of Mathematics and Physics, Fujian University of Technology, Fuzhou, Fujian 350108, China

Received 7 December 2009; Accepted 15 January 2010

Academic Editor: Leonid Shaikhet

Copyright © 2010 Tieshan He et al. 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 second-order three-point discrete boundary value problem. By using the topological degree theory and the fixed point index theory, we provide sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions. As an application, an example is given to demonstrate our main results.

1. Introduction

In this paper, we consider the following second-order three-point discrete boundary value problem (BVP):

where , is the discrete interval , , , , , and is a continuous function.

Boundary value problems for difference equations arise in different areas of applied mathematics and physics. Existence and multiplicity of positive solutions or nontrivial solutions for boundary value problems of difference equations have been extensively studied in the literature; see [19] and the references therein.

On the other hand, in the existing literature, there are some papers studying the sign-changing solutions for boundary value problems of differential equations; for example, see [1012]. But the problems of the existence of sign-changing solutions to discrete multipoint boundary value problems have received very little attention in the literature to the best knowledge of the authors. In this paper, motivated by [12, 13], we aim to study the existence of multiple sign-changing solutions to the second-order three-point discrete boundary value problem (1.1). Under some suitable conditions, we prove that the three-point discrete boundary value problem (1.1) has at least two sign-changing solutions, two positive solutions, and two negative solutions. The main approach is the topological degree theory and the fixed point index theory.

The organization of this paper is as follows. In Section 2, we present some preliminary knowledge about the topological degree theory and the fixed point index theory and use the knowledge to obtain some lemmas which are very crucial in our main results. In Section 3, by computing the topological degree and the fixed point index, we discuss the existence of multiple sign-changing solutions to BVP (1.1), and a simple example is given.

2. Preliminaries

As we have mentioned, we will use the theory of the Leray-Schauder degree and the fixed point index in a cone to prove our main existence results. Let us collect some results that will be used below. One can refer to [1316] for more details.

Lemma 2.1 (see [13, 14]). Let be a Banach space and, be a cone in . Assume that is a bounded open subset of . Suppose that is a completely continuous operator. If there exists such that then the fixed point index .

Lemma 2.2 (see [13, 14]). Let be a Banach space and let be a cone in . Assume that is a bounded open subset of , . Suppose that is a completely continuous operator. If then the fixed point index .

Lemma 2.3 (see [15]). Let be a Banach space, let be a bounded open subset of , , and be completely continuous. Suppose that then .

Lemma 2.4 (see [16]). Let be a completely continuous operator which is defined on a Banach space . Let be a fixed point of and assume that is defined in a neighborhood of and Fréchet differentiable at . If 1 is not an eigenvalue of the linear operator , then is an isolated singular point of the completely continuous vector field and for small enough , where is the sum of the algebraic multiplicities of the real eigenvalues of in .

Lemma 2.5 (see [16]). Let be a completely continuous operator which is defined on a Banach space . Assume that 1 is not an eigenvalue of the asymptotic derivative. Then the completely continuous vector field is nonsingular on spheres of sufficiently large radius and where is the sum of the algebraic multiplicities of the real eigenvalues of in .

From [12, Lemma 2.4], we have the following lemma.

Lemma 2.6. Let be a solid cone of a Banach space ( is nonempty), let be a relatively bounded open subset of , and let be a completely continuous operator. If any fixed point of in is an interior point of , there exists an open subset of () such that

Now we shall consider the space

equipped with the norm . Clearly is a dimensional Banach space. Choose the cone defined by

Obviously, the interior of is . For each , we write if for . A solution of BVP (1.1) is said to be a positive solution (a negative solution, resp.) if (, resp.). A solution of BVP (1.1) is said to be a sign-changing solution if .

Lemma 2.7. Let be fixed. Then the problem has a unique solution where is given by

Proof. We use a similar approach to that in [17, Lemmas , ]. From , we have Suming the above equations, one gets where, and in what follows, we denote when . Again summing (2.13) from to , it follows that where . Since and , one gets By (2.14) and (2.15), we have When , it follows from (2.16) that When , it follows from (2.16) that Then, the unique solution of (2.9) can be written as .

Remark 2.8. Green’s function defined by Lemma 2.7 is positive on .
Define operators , respectively, by
Now from Lemma 2.7, it is easy to see that BVP (1.1) has a solution if and only if is a fixed point of the operator . It follows from the continuity of that is completely continuous.
We shall use the following assumptions.
()We have , or where , and denotes the integer part of the real number . ()For any , ; for any and , ()There exists an even number such that where uniformly for , and , are given in Lemma 2.9, ()There exists an even number such that where uniformly for , and are given in condition ().()There exists a constant such that for any , where .

Lemma 2.9. Suppose that () holds; then there exist with such that , .

Proof. First, suppose that . Let , then we have It follows from the intermediate value theorem that there exist and , such that , .
Now suppose that and . Let . It is easy to see that there exist with such that . Then Lemma 2.9 is proved.

Remark 2.10. Condition is reasonable. For example, let , , then . Let , , then .

Lemma 2.11. Suppose that () holds; then the set of eigenvalues of the linear operator consists of the strictly decreasing finite sequence of , with corresponding eigenfunctions , where , , and are given in Lemma 2.9. In addition, the algebraic multiplicity of each eigenvalue of the linear operator is equal to .

Proof. It is easy to see that is equivalent to the following equation: By Lemma 2.9, we suppose that is a nontrivial solution of (2.28). Then, Hence, for any , is an eigenvalue of the linear operator with the corresponding eigenfunction . Since the linear operator is identified with a linear transformation from to , the set of eigenvalues of the linear operator consists of the strictly decreasing finite sequence of . Obviously, the algebraic multiplicity of each eigenvalue of is equal to . This completes the proof.

Remark 2.12. When , we see that BVP (1.1) is reduced to Dirichlet boundary value problem and , . When and , BVP (1.1) is reduced to the focal boundary value problem and , .

Lemma 2.13. Suppose that () holds, and is a solution of BVP (1.1). Then .

Proof. If for some , then So , and it follows that if is zero somewhere in then it vanishes identically in .

Remark 2.14. Similarly to Lemma 2.13, we know also that if () holds and is a solution of BVP (1.1), then .

Lemma 2.15. Suppose that ()–() hold. Then the operator is Fréchet differentiable at and , where operator is defined by (2.20). Moreover, and .

Proof. By (), for any , there exist such that for any , . Hence, noticing that for any , we have for any with , where . Consequently, This means that the nonlinear operator is Fréchet differentiable at , and .
By (), for any , there exist such that for any , . Let . By the continuity of with respect to , we have . Then, for any , . Thus
for any . Consequently, which implies that operator is Fréchet differentiable at , and . The proof is completed.

Lemma 2.16. Let be given in condition (). Suppose that ()–() hold. Then, , . Moreover, one has the following.(i)There exists an such that for any , (ii)There exists an such that for any ,

Proof. By () and the fact that is positive on , we get that for any , , , and , . Then and .
We only need to prove conclusion (i). The proof of conclusion (ii) is similar and will be omitted here. Let . Condition () yields . It follows from (2.32) that there exists such that
where Setting then is completely continuous. For any and , we obtain that According to the homotopy invariance of the fixed point index, for any , we have Let . Then and (see Lemma 2.11 and the proof of Lemma 2.9). We claim Indeed, we assume that there exist and such that . Obviously, . Since , then . Set . It is clear that and . Then Since , then , which contradicts with the definition of . This proves (2.41).
It follows from Lemma 2.1 and (2.41) that
Similarly to (2.43), we know also that By (2.39), (2.43), (2.40), and (2.44), we conclude

3. Main Results

Now with the aid of the lemmas in Section 2, we are in position to state and prove our main results.

Theorem 3.1. Assume that the conditions ()–() hold. Then BVP (1.1) has at least two sign-changing solutions. Moreover, BVP (1.1) has at least two positive solutions and two negative solutions.

Proof. From the proof of Lemma 2.7, we have Since is positive on , by (), we have for any with , This gives By (3.3) and Lemmas 2.3 and 2.2, we have From () and Lemma 2.11, one has that the eigenvalues of the operator which are larger than 1 are From () and Lemma 2.11, one has that the eigenvalues of the operator which are larger than 1 are It follows from Lemmas 2.4 and 2.5 that there exist and such that where and are given in Lemma 2.16. Owing to Lemma 2.16, one has According to the additivity of the fixed point index, by (3.5), (3.11), and (3.13), we have Hence, the nonlinear operator has at least two fixed points and , respectively. Then, and are positive solutions of BVP (1.1).
Using again the additivity of the fixed point index, by (3.6), (3.12), and (3.14), we get
Hence, the nonlinear operator has at least two fixed points and , respectively. Then, and are negative solutions of BVP (1.1). Let It follows from Lemmas 2.6, 2.13, Remark 2.14, and (3.15)–(3.18) that there exist open subsets and of such that By (3.4), (3.21), (3.23), (3.9), and the additivity of Leray-Schauder degree, we get which implies that the nonlinear operator has at least one fixed point .
Similarly, by (3.10), (3.22), (3.24), and (3.4), we get
which implies that the nonlinear operator has at least one fixed point . Then, and are two distinct sign-changing solutions of BVP (1.1). Thus, the proof of Theorem 3.1 is finished.

Theorem 3.2. Assume that the conditions ()–() hold, and that for and . Then BVP (1.1) has at least four sign-changing solutions. Moreover, BVP (1.1) has at least two positive solutions and two negative solutions.

Proof. It follows from the proof of Theorem 3.1 that BVP (1.1) has at least six different nontrivial solutions satisfying that By the condition that for and , we know that and are also solutions of BVP (1.1). Let , , then are different nontrivial solutions of BVP (1.1). The proof is completed.

By the method used in the proof of Theorems 3.1 and 3.2, we can prove the following corollaries.

Corollary 3.3. Assume that the conditions ()–() and () or (), (), (), and () hold. Then BVP (1.1) has at least one sign-changing solution. Moreover, BVP (1.1) has at least one positive solution and one negative solution.

Corollary 3.4. Assume that the conditions ()–() and () or (), (), (), and () hold, and that for and . Then BVP (1.1) has at least two sign-changing solutions. Moreover, BVP (1.1) has at least one positive solution and one negative solution.

Next, we present a simple example to which Theorem 3.2 can be applied.

Example 3.5. Consider the second-order three-point discrete boundary value problem where ,  ,   and . Obviously, and for all .
From Lemma 2.11 and the proof of Lemma 2.9, we know that the set of eigenvalues of the linear operator (see (2.19)) consists of the strictly decreasing finite sequence of , , where Then the conditions ()–() hold. Since for all , then () holds with and . Therefore, by Theorem 3.2, BVP (3.28) has at least four sign-changing solutions. Moreover, BVP (3.28) has at least two positive solutions and two negative solutions.

Acknowledgment

This project is supported by the Education Department of Fujian Province of China (no. JA06035).

References

  1. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. View at Zentralblatt MATH · View at MathSciNet
  2. R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. View at Zentralblatt MATH · View at MathSciNet
  3. R. P. Agarwal, K. Perera, and D. O'Regan, “Multiple positive solutions of singular and nonsingular discrete problems via variational methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no. 1-2, pp. 69–73, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. Yu and Z. Guo, “On boundary value problems for a discrete generalized Emden-Fowler equation,” Journal of Differential Equations, vol. 231, no. 1, pp. 18–31, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. G. Bonanno and P. Candito, “Nonlinear difference equations investigated via critical point methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3180–3186, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. Zhang and Z. Yang, “Positive solutions of a general discrete boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 469–481, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. Bai and Y. Xu, “Nontrivial solutions of boundary value problems of second-order difference equations,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 297–302, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. L. Jiang and Z. Zhou, “Existence 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. D. Wang and W. Guan, “Three positive solutions of boundary value problems for p-Laplacian difference equations,” Computers & Mathematics with Applications, vol. 55, no. 9, pp. 1943–1949, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  10. X. Xu and J. Sun, “On sign-changing solution for some three-point boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 4, pp. 491–505, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. X. Xu and D. O'Regan, “Multiplicity of sign-changing solutions for some four-point boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 2, pp. 434–447, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. C. Pang, W. Dong, and Z. Wei, “Multiple solutions for fourth-order boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 464–476, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. J. Guo, J. X. Sun, and Z. L. Liu, Functional Methods for Nonlinear Differential Equations, Shandong Science and Technology Press, Ji'nan, China, 1995.
  14. D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988. View at MathSciNet
  15. D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Ji'nan, China, 2nd edition, 2001.
  16. M. A. Krasnoselskiĭ and P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, vol. 263 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1984. View at MathSciNet
  17. Y. Guo and W. Ge, “Positive solutions for three-point boundary value problems with dependence on the first order derivative,” Journal of Mathematical Analysis and Applications, vol. 290, no. 1, pp. 291–301, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet