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Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 143040, 16 pages
http://dx.doi.org/10.1155/2008/143040
Research Article

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

School of Mathematics and Information Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 7 August 2008; Revised 8 October 2008; Accepted 9 November 2008

Academic Editor: Leonid Shaikhet

Copyright © 2008 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 -Laplacian -point boundary value problems on time scales: , , , , , where is -Laplacian operator, that is, , , . We obtain the existence of 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 means of unifying and extending theories from differential and difference equations. The study of time scales has lead 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 [713].

A time scale is a nonempty closed subset of We make the blanket assumption that 0, T are points in . By an interval (0, T), we always mean the intersection of the real interval (0, T) with the given time scale; that is (0, T).

In [14], Anderson considered the the following third-order nonlinear boundary value problem (BVP): Author studied the existence of solutions for the nonlinear boundary value problem by using the Krasnoselskii's fixed point theorem and Leggett and Williams fixed point theorem, respectively.

In [8, 9], He considered the existence of positive solutions of the -Laplacian dynamic equations on time scalessatisfying the boundary conditionsorwhere . He obtained the existence of at least double and triple positive solutions of the boundary value problems by using a new double fixed point theorem and triple fixed point theorem, respectively.

In [13], 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 third-order -Laplacian -point boundary value problems on time scales. This paper attempts to fill this gap in the literature.

In this paper, by using different method, we are concerned with the existence of positive solutions for the following third-order -Laplacian -point boundary value problems on time scales:where is -Laplacian operator, that is, and satisfy

(H1), , , (H2) is continuous, and there exists such that , where .

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; if , is said to be left scattered; if , is said to be right dense; 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 thatfor 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 thatfor 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 one defines the delta integral byIf , then one defines the nabla integral by

Lemma 2.5. If then for the boundary value problem (BVP) has the unique solution

Proof. By direct computation, we can easily get (2.7). So, we omit it.

Lemma 2.6. If then for the boundary value problem (BVP) has the unique solutionwhere .

Proof. Integrating both sides of (1.6) on , we haveSoBy boundary value condition , we haveBy (2.10) and (2.12), we knowThis together with Lemma 2.5 implies thatwhere . The proof is complete.

Lemma 2.7. Let If and then the unique solution of (2.8) satisfies

Proof. By , we can know that the graph of is concave down on . So we only prove
Firstly, we shall prove by the following two perspectives.
(i) If , we have(ii)If , by (2.8), we haveOn the other hand, we haveThe proof is completed.

Lemma 2.8. Let If and then the unique positive solution of (BVP) (2.8) satisfies where .

Proof. Let , we shall discuss it from the following two perspectives.
Case 1. If .
Firstly, assume , then By we haveSoSecondly, assume , then . Otherwise, we have , then . By we havea contradiction.
By concave of , we get . In fact, since , then , which implies

Case 2. If .
Firstly, assume , then . By concave of we have , which implies , then
Secondly, assume , then , and . If not, , then . So, we havea contradiction. By there exists such that , then . By concave of we have , thenTherefore, by (2.21)–(2.26), we havewhere . The proof is complete.

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 byDefine an operator by settingwhere . Obviously, is a solution of boundary value problem (1.6) if and only if is a fixed point of operator .

Lemma 2.9. is completely continuous.

Proof. By and Lemmas 2.7-2.8, we easily get . By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove is completely continuous.

Lemma 2.10 (see [15]). Let be a cone in a Banach space Let be an open bounded subset of with and Assume that is a compact map such that for Then the following results hold.
(1)If , , then .(2)If there exists such that for all and all , then .(3)Let be open in such that . If and , then has a fixed point in . The same result holds if and , where denotes fixed point index.
We define

Lemma 2.11 (see [15]). defined above has the following properties:
(a)(b) is open relative to K;(c) if and only if (d)if , then for .
For the convenience, we introduce the following notations:

Lemma 2.12. If satisfies the following condition: then

Proof. For , then from (2.32), we haveSo thatTherefore,This implies that for . Hence, by Lemma 2.10(1) it follows that .

Lemma 2.13. If satisfies the following condition: then

Proof. Let for . Then . We claim thatIn fact, if not, there exist and such that . By , we haveSo thatBy [16, Theorem 2.2(iv)], for we haveSo, for we haveTherefore,Obviously, we can knowFor , thenThis together with Lemma 2.11(c) implies thata contradiction. Hence, by Lemma 2.10(2), it follows that .

3. Main Results

We now give our results on the existence of positive solutions of BVP (1.6).

Theorem 3.1. Suppose conditions and hold, and assume that one of the following conditions hold.
(H3) There exist with such that .(H4) There exist with such that . Then, the boundary value problem (1.6) has at least one positive solution.

Proof. Assume that holds, we show that has a fixed point in . By and Lemma 2.12, we have thatBy and Lemma 2.13, we have thatBy Lemma 2.11(a) and , we have . It follows from Lemma 2.10(3) that has a fixed point in When condition holds, the proof is similar to the above, so we omit it here.
As a special case of Theorem 3.1, we obtain the following result.

Corollary 3.2. Suppose conditions , hold, and assume that one of the following conditions holds.
(H5) and .(H6) and . Then, the boundary value problem (1.6) has at least one positive solution.

Theorem 3.3. Assume conditions and hold, and suppose that one of the following conditions holds.
(H7) There exist , and with and such that (H8) There exist , and with such that Then, the boundary value problem (1.6) has at least two positive solutions. Moreover, if in is replaced by then the BVP (1.6) has a third positive solution .

Proof. Assume that condition holds, we show that either has a fixed point in or If for By Lemma 2.12 and Lemma 2.13, we have that By Lemma 2.11(a) and , we have It follows from Lemma 2.10(3) that has a fixed point in Similarly, has a fixed point in When condition holds, the proof is similar to the above, so we omit it here.
As a special case of Theorem 3.3, we obtain the following result.

Corollary 3.4. Assume conditions and hold, if there exists such that one of the following conditions holds.
(H9) and .(H10) and . Then, the boundary value problem (1.6) has at least two positive solutions.

4. Some Examples

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

Example 4.1. Consider the following three-point boundary value problem with -Laplacian:where . By computing, we can know . Let , then . We define a nonlinearity as follows:Then, by the definition of , we have
(i);(ii).
So condition holds, by Theorem 3.1, boundary value problem (4.1) has at least one positive solution.

Acknowledgment

Project supported by the National Natural Science Foundation of China (10471075).

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