Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 123283, 15 pages
http://dx.doi.org/10.1155/2009/123283
Research Article

Triple Positive Solutions for Third-Order -Point Boundary Value Problems on Time Scales

1School of Statistics and Mathematics Science, Shandong Economics University, Jinan, Shandong 250014, China
2School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 25 February 2009; Accepted 25 May 2009

Academic Editor: Leonid Shaikhet

Copyright © 2009 Jian Liu and Fuyi Xu. 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 study the following third-order -point boundary value problems on time scales , , , , , where is an increasing homeomorphism and homomorphism and , . We obtain the existence of three positive solutions by using fixed-point theorem in cones. The conclusions in this paper essentially extend and improve the known results.

1. Introduction

The theory of time scales was initiated by Hilger [1] as a mean of unifying and extending theories from differential and difference equations. The study of time scales has led to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models; see, for example [26]. Recently, the boundary value problems with -Laplacian operator have also been discussed extensively in the literature, for example, see [715].

A time scale is a nonempty closed subset of . We make the blanket assumption that are points in . By an interval , we always mean the intersection of the real interval with the given time scale; that is, .

In [16], Anderson considered the following third-order nonlinear boundary value problem (BVP): He used the Krasnoselskii and Leggett-Williams fixed-point theorems to prove the existence of solutions to the nonlinear boundary value problem.

In [9, 10], He considered the existence of positive solutions of the -Laplacian dynamic equations on time scales satisfying the boundary conditions or where . He obtained the existence of at least double and triple positive solutions of the problems by using a new double fixed point theorem and triple fixed point theorem, respectively.

In [15], Zhou and Ma firstly studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with -Laplacian operator They established a corresponding iterative scheme for the problem by using the monotone iterative technique.

However, to the best of our knowledge, little work has been done on the existence of positive solutions for the increasing homeomorphism and positive homomorphism operator on time scales. So the goal of the present paper is to improve and generate -Laplacian operator and establish some criteria for the existence of multiple positive solutions for the following third-order -point boundary value problems on time scales where is an increasing homeomorphism and homomorphism and , and satisfy

(), , () is continuous, and there exits such that , where .

A projection is called an increasing homeomorphism and homomorphism, if the following conditions are satisfied:

(i)if , then ;(ii) is continuous bijection and its inverse mapping is also continuous;(iii)

2. Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in [15].

Definition 2.1. A time scale is a nonempty closed subset of real numbers . For and , define the forward jump operator and backward jump operator , respectively, by for all . If , is said to be right scattered, and if , is said to be left scattered; if , is said to be right dense, and if , is said to be left dense. If has a right scattered minimum , define ; otherwise set . If has a left scattered maximum , define ; otherwise set .

Definition 2.2. For and , the delta derivative of at the point is defined to be the number (provided it exists) with the property that for each , there is a neighborhood of such that for all .
For and , the nabla derivative of at , denoted by (provided it exists) with the property that for each , there is a neighborhood of such that for all .

Definition 2.3. A function is left-dense continuous (i.e., -continuous), if is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in .

Definition 2.4. If , then we define the delta integral by If , then we define the nabla integral by

Definition 2.5. Let be a real Banach space over . A nonempty closed set is said to be a cone provided that
(i), implies ; (ii) implies .

Definition 2.6. Given a cone in a real Banach space , a functional is said to be increasing on , provided , for all with .

Definition 2.7. Given a cone in a real Banach space , we define for each the set

Definition 2.8. A map is called nonnegative continuous concave functional on a cone of a real Banach space if is continuous and for all and . Similarly we say that the map is called nonnegative continuous concave functional on a cone of a real Banach space if is continuous and for all and .

Let be nonnegative continuous convex functionals on , let be a nonnegative continuous concave functional on , and let be a nonnegative continuous functional on . For nonnegative real numbers , and we define the following convex set:

Theorem 2.9 ([17]). Let be a cone in a real Banach space . Let and be nonnegative continuous convex functionals on , let be a nonnegative continuous concave functional on , and let be a nonnegative continuous functional on satisfying for , such that for some positive numbers and , and for all . Suppose that is a completely continuous operator and there exist nonnegative numbers , and with such that
(i) and for ;(ii) for with ;(iii) and for with .Then has at least three fixed points satisfying

Theorem 2.10 ([18]). Let be a bounded closed convex subset of a Banach space . Assume that are disjoint closed convex subsets of and are nonempty open subsets of with and . Suppose that is completely continuous and the following conditions hold:
(i);(ii) has no fixed points in .Then has at least three points such that , and .

Lemma 2.11. If condition holds, then for , the boundary value problem (BVP) has the unique solution

Proof. By caculating, we can easily get (2.12). So we omit it.

Lemma 2.12. If condition holds, then for , the boundary value problem (BVP) has the unique solution where , is the inverse function to .

Proof. Integrating both sides of equation in (2.13) on , we have So, By boundary value condition , we have By (2.15) and (2.17) we know This together with Lemma 2.11 implies that where . The proof is complete.

Lemma 2.13. Let condition hold. If and , then the unique solution of (2.13) satisfies

Proof. By , we can know that the graph of is concave down on and is nonincreasing on . This together with the assumption that the boundary condition is implies that for . This implies that So we only prove By condition we have The proof is completed.

3. Triple Positive Solutions

In this section, some existence results of positive solutions to BVP (1.6) are established by imposing some conditions on and defining a suitable Banach space and a cone.

Let be endowed with the ordering if for all and is defined as usual by maximum norm. Clearly, it follows that is a Banach space.

We define a cone by Let and fix such that and define the nonnegative continuous convex functionals and , the nonnegative continuous concave functional , and the nonnegative continuous functional on the cone by For notational convenience, denote

Lemma 3.1 ([9]). If , then
(1) for all ;(2) for with .
Define an operator by where . Then, is a solution of boundary value problem (1.6) if and only if is a fixed point of operator . Obviously, for one has for . In addition, for and , this implies . With standard argument one may show that is completely continuous.

Theorem 3.2. Suppose conditions and hold, and there exist positive numbers such that
();();().Then, the BVP (1.6) has at least three positive solutions satisfying

Proof. Based on Lemma 3.1, it is clear that for and , there are and . Furthermore, and therefore .
Take , then . By means of () one derives Thus .
Set and , it follows that which means
For , we have for . By condition () we have So, (i) of Theorem 2.9 is fulfilled.
If and , then due to of Lemma 3.1 Therefore, (ii) of Theorem 2.9 is fulfilled.
Take and , then , it then follows from () that As a result, all the conditions of Theorem 2.9 are verified. This completes the proof.

Theorem 3.3. Suppose that conditions () and () hold. Let and assume that the following conditions are satisfied:
(), ;() there exists a number such that , ;(), .Then, the BVP (1.6) has at least three positive solutions , and such that Where for real number , is continuous, , for .

Proof. We first show that if condition () holds. If , then , which implies . We have This implies that .
Next, condition () indicates that there exists such that . Now we let where is the interior of . Then we have . Moreover, means . Thus has no fixed point in .
To show and has no fixed point in , set , following the definition of , we can know , for . Condition () then gives rise to , which in turn produces Combining the above two inequalities one achieves , for . That is, . So and has no fixed point in . Therefore, all conditions of Theorem 2.10 are fulfilled, and the BVP (1.6) has at least three positive solutions , and such that

4. Some Examples

In the section, we present some simple examples to explain our results. We only study the case .

Example 4.1. Consider the following third-order three-point boundary value problem: where .

We choose , by computing we can know . Let , then . Obviously, . We define a nonlinearity as follows: Then, by the definition of , we have

(i);(ii);(iii).

By Theorem 3.2, BVP (4.1) has at least three positive solutions.

Example 4.2. Consider the following third-order three-point boundary value problem: where .

By computing, we can know . Let , then . Obviously, . We define a nonlinearity as follows: Then, by the definition of , we have

(i);(ii)and there exists such that ;(iii).

By Theorem 3.3, BVP (4.3) has at least three positive solutions.

Remark 4.3. Consider following nonlinear m-point boundary value problem: where and satisfy the conditions () and (). It is clear that is an increasing homeomorphism and homomorphism and . Because -Laplacian operators are odd, they do not apply to our example. Hence we generalize boundary value problem with -Laplacian operator, and the results [811, 1315] do not apply to the example.

Remark 4.4. In a similar way, we can get the corresponding results for the following boundary value problem:

Acknowledgment

The project was supported by the National Natural Science Foundation of China (10471075).

References

  1. S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. R. P. Agarwal and D. O'Regan, “Nonlinear boundary value problems on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 44, no. 4, pp. 527–535, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. F. M. Atici and G. Sh. Guseinov, “On Green's functions and positive solutions for boundary value problems on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 75–99, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. H.-R. Sun and W.-T. Li, “Positive solutions for nonlinear three-point boundary value problems on time scales,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 508–524, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Bohner and A. Peterso, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. View at MathSciNet
  6. H. R. Sun and W.-T. Li, “Positive solutions for nonlinear m-point boundary value problems on time scales,” Acta Mathematica Sinica, vol. 49, no. 2, pp. 369–380, 2006 (Chinese). View at Google Scholar · View at MathSciNet
  7. H.-R. Sun and W.-T. Li, “Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales,” Journal of Differential Equations, vol. 240, no. 2, pp. 217–248, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  8. Y.-H. Su, W.-T. Li, and H.-R. Sun, “Triple positive pseudo-symmetric solutions of three-point BVPs for p-Laplacian dynamic equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1442–1452, 2008. View at Google Scholar · View at MathSciNet
  9. Z. He, “Double positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 182, no. 2, pp. 304–315, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Z. He and X. Jiang, “Triple positive solutions of boundary value problems for p-Laplacian dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 911–920, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  11. D. Anderson, R. Avery, and J. Henderson, “Existence of solutions for a one dimensional p-Laplacian on time-scales,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 889–896, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  12. D.-X. Ma, Z.-J. Du, and W.-G. Ge, “Existence and iteration of monotone positive solutions for multipoint boundary value problem with p-Laplacian operator,” Computers & Mathematics with Applications, vol. 50, no. 5-6, pp. 729–739, 2005. View at Google Scholar · View at MathSciNet
  13. Y. Wang and W. Ge, “Existence of multiple positive solutions for multipoint boundary value problems with a one-dimensional p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 2, pp. 476–485, 2007. View at Publisher · View at Google Scholar
  14. Y. Wang and C. Hou, “Existence of multiple positive solutions for one-dimensional p-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 144–153, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  15. C. Zhou and D. Ma, “Existence and iteration of positive solutions for a generalized right-focal boundary value problem with p-Laplacian operator,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 409–424, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  16. D. R. Anderson, “Green's function for a third-order generalized right focal problem,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 1–14, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. Liu and W. Ge, “Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 551–561, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  18. 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