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Abstract and Applied Analysis
Volume 2010, Article ID 261741, 19 pages
http://dx.doi.org/10.1155/2010/261741
Research Article

Multiple Positive Solutions of a Second Order Nonlinear Semipositone -Point Boundary Value Problem on Time Scales

1Department of Mathematics, East China Normal University, Shanghai 200241, China
2School of Mathematics and Computer, Harbin University, Harbin, Heilongjiang 150086, China

Received 28 April 2010; Revised 26 June 2010; Accepted 7 September 2010

Academic Editor: Allan C Peterson

Copyright © 2010 Chengjun Yuan and Yongming Liu. 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

In this paper, we study a general second-order -point boundary value problem for nonlinear singular dynamic equation on time scales , , , . This paper shows the existence of multiple positive solutions if is semipositone and superlinear. The arguments are based upon fixed-point theorems in a cone.

1. Introduction

In this paper, we consider the following dynamic equation on time scales: where ; for all ; and satisfy is continuously and nonnegative function and there exists s.t. , may be singular at ;, .

In the past few years, the boundary value problems of dynamic equations on time scales have been studied by many authors (see [115] and references therein). Recently, multiple-point boundary value problems on time scale have been studied, for instance, see [19].

In 2008, Lin and Du [2] studied the -point boundary value problem for second-order dynamic equations on time scales: where is a time scale. This paper deals with the existence of multiple positive solutions for second-order dynamic equations on time scales. By using Green's function and the Leggett-Williams fixed point theorem in an appropriate cone, the existence of at least three positive solutions of the problem is obtained.

In 2009, Topal and Yantir [1] studied the general second-order nonlinear -point boundary value problems (1.1) with no singularities and the case. The authors deal with the determining the value of ; the existences of multiple positive solutions of (1.1) are obtained by using the Krasnosel'skii and Legget-William fixed point theorems.

Motivated by the abovementioned results, we continue to study the general second-order nonlinear -point boundary value problem (1.1), but the nonlinear term may be singularity and semipositone.

In this paper, the nonlinear term of (1.1) is suit to and semipositone and the superlinear case, we will prove our two existence results for problem (1.1) by using Krasnosel'skii fixed point theorem. This paper is organized as follows. In Section 2, starting with some preliminary lemmas, we state the Krasnosel'skii fixed point theorem. In Section 3, we give the main result which state the sufficient conditions for the -point boundary value problem (1.1) to have existence of positive solutions.

2. Preliminaries

In this section, we state the preliminary information that we need to prove the main results. From Lemmas and in [1], we have the following lemma.

Lemma 2.1 (see [1]). Assuming that (C2) holds. Then the equations have unique solutions and , respectively, and(a) is strictly increasing on ,(b) is strictly decreasing on .

For the rest of the paper we need the following assumption:(C3).

Lemma 2.2 (see [1]). Assuming that (C2) and (C3) hold. Let . Then boundary value problem is equivalent to integral equation where

Proof. First we show that the unique solution of (2.3) can be represented by (2.4). From Lemma 2.1, we know that the homogenous part of (2.3) has two linearly independent solution and since
Now by the method of variations of constants, we can obtain the unique solution of (2.3) which can be represented by (2.4) where and are as in (2.5) and (2.6), respectively. Next we check the function defined in (2.4) is the solution of the boundary value problem (2.3). For this purpose we first show that (2.4) satisfies (2.3). From the definition of Green's function (2.6), we get Hence, the derivatives and are as follows: Replacing the derivatives in (2.3), we deduce that Therefore the function defined in (2.4) satisfies (2.3). Further we obtain that the boundary value conditions are satisfied by (2.4). The first condition follows from (2.5) and (2.6) and Lemma 2.1. Now we verify the second boundary condition. Since we obtain that On the other hand, by using (2.5), we find that Combining the two equations above finishes the proof.

Lemma 2.3. Green's function has the following properties:

Lemma 2.4. Assume that (C2) and (C3) hold. Let be a solution of boundary value problem (1.1) if and only if is a solution of the following integral equation: where

The proofs of the Lemmas 2.3 and 2.4 can be obtained easily by Lemmas 2.1 and 2.2.

Lemma 2.5. Green's function defined by (2.16) has the following properties: where

Proof. From Lemma 2.3, we have The proof is complete.

The following theorems will play major role in our next analysis.

Theorem 2.6 (see [16]). Let be a Banach space, and let be a cone in . Let be open subsets of with , and let be a completely continuous operator, such that, either (1), , , , or (2), , . Then has a fixed point in .

3. Main Results

We make the following assumptions:(H), moreover there exists a function such that , for any , .(H) may be singular at , moreover there exists a function such that , for any , .(H), for .(H) There exists such that (H) for any , is any constant.

In fact, we only consider the boundary value problem where and , which is the solution of the boundary value problem From Lemma 2.1, it is easy to verify that and .

We will show that there exists a solution for boundary value problem (3.1) with . If this is true, then is a nonnegative solution (positive on ) of boundary value problem (3.1). Since for any , from we have As a result, we will concentrate our study on boundary value problem (3.1).

We note that is a solution of (3.1) if and only if

For our constructions, we will consider the Banach space equipped with standard norm . We define a cone by where is defined by Lemma 2.1 (namely, is solution (2.1)). Define an integral operator by Notice, from (3.8) and Lemma 2.5, we have on for and then .

On the other hand, we have Thus, . In addition, standard arguments show that and is a compact, and completely continuous.

Theorem 3.1. Suppose that (H)-(H) hold. Then there exists a constant such that, for any , boundary value problem (1.1) has at least one positive solution.

Proof. Fix . From (H), let be such that Suppose that where and . Since there exists a such that
Let and be such that , we claim that . In fact that is, which implies that . Let . By nonlinear alternative of Leray-Schauder type theorem, has a fixed point . Moreover, combing (3.8), (3.28), and , we obtain that Let Then (1.1) has a positive solution and . This completes the proof of Theorem 3.1.

Theorem 3.2. Suppose that (H) and (H)-(H) hold. Then there exists a constant such that, for any , boundary value problem (1.1) has at least one positive solution.

Proof. Let , where and . Choose Then for any , then and , we have This implies that
On the other hand, choose a constant such that By assumption (H), for any , there exists a constant such that Choose , and let , then for any , we have Then, Now,
Condition (2.1) of Krasnosel'skii’s fixed-point theorem is satisfied. So has a fixed point with such that Since , Let , then is a positive solution of boundary value problem (1.1). This completes the proof of Theorem 3.2.

Since condition (H) implies conditions (H) and (H), and from proof of Theorems 3.1 and 3.2, we immediately have the following theorem.

Theorem 3.3. Suppose that (H)–(H) hold. Then boundary value problem (1.1) has at least two positive solutions for sufficiently small.

Proof. On the hand, fix . From (H), let be such that Choose where , , and .
On the other hand, set , where . Choose
So, let
From , we have , from proof of Theorem 3.1, we know that (1.1) has a positive solution and . Further, also from , we have , from proof of Theorem 3.2, we know that (1.1) has a positive solution and . Then (1.1) has at least two positive solutions and . This completes the proof of Theorem 3.3.

4. Examples

To illustrate the usefulness of the results, we give some examples.

Example 4.1. Consider the boundary value problem where . Then, if is sufficiently small, (4.1) has a positive solution with for
To see this, we will apply Theorem 3.2 with Clearly for . Namely, (H) and (H)–(H) hold. From , set and , we have and . Now, if , Theorem 3.2 guarantees that (4.1) has a positive solution with .

Example 4.2. Consider the boundary value problem Then, if is sufficiently small, (4.5) has two solutions with for
To see this, we will apply Theorem 3.3 with Clearly Namely, (H)–(H) hold. Let , and , then we may have Now, if , Theorem 3.2 guarantees that (4.5) has a positive solution with .
Next, let , where is a constant, then we have and . Now, if , Theorem 3.1 guarantees that (4.5) has a positive solution with .
So, since all the conditions of Theorem 3.3 are satisfied, if , Theorem 3.3 guarantees that (4.5) has two solutions with   

Example 4.3. Consider the boundary value problem where . Then, if is sufficiently small, (4.10) has two solutions with for
To see this we will apply Theorem 3.3 with Clearly Namely, (H)–(H) hold. Let , and , then we may have Now, if then , Theorem 3.2 guarantees that (4.10) has a positive solution with .
Next, let , where is a constant, then we have and . Now, if then , Theorem 3.1 guarantees that (4.10) has a positive solution with .
So, if , Theorem 3.3 guarantees that (4.10) has two solutions with   

Example 4.4. Let . We consider the following four point boundary value problem: where , and . Then, if is sufficiently small, (4.15) has two solutions with ().
Let and be the solutions of the following linear boundary value problems, respectively, It is evident (form the Corollaries 4.24 and 4.25 and Theorem of [17]) that Also satisfies (C3). Green's function is and follows from on .
To see this, we will apply Theorem 3.3 with Clearly Namely, (H)–(H) hold. Let , and , then we may have Now, if then , Theorem 3.2 guarantees that (4.15) has a positive solution with .
Next, let is a constant, then we have and . Now, if then , Theorem 3.1 guarantees that (4.15) has a positive solution with .
So, if , Theorem 3.3 guarantees that (4.15) has two solutions with   

Acknowledgments

The work was supported by the Scientific Research Fund of Heilongjiang Provincial Education Department (no. 11544032) and the National Natural Science Foundation of China (no. 10871071).

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