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Abstract and Applied Analysis
Volume 2009, Article ID 341679, 16 pages
http://dx.doi.org/10.1155/2009/341679
Research Article

Solvability of a Higher-Order Three-Point Boundary Value Problem on Time Scales

Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Received 3 August 2009; Revised 17 October 2009; Accepted 19 November 2009

Academic Editor: Ferhan Atici

Copyright © 2009 Yanbin Sang. 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 a higher-order three-point boundary value problem on time scales. A new existence result is first obtained by using a fixed point theorem due to Krasnoselskii and Zabreiko. Later, under certain growth conditions imposed on the nonlinearity, several sufficient conditions for the existence of a nonnegative and nontrivial solution are obtained by using Leray-Schauder nonlinear alternative. Our conditions imposed on nonlinearity are all very easy to verify; as an application, some examples to demonstrate our results are given.

1. Introduction

We are concerned with the following even-order three-point boundary value problem on time scales : ; we assume that is right dense so that for and that for each , coefficients satisfy the following condition:

Throughout this paper, we let be any time scale and let be a subset of such that . Some preliminary definitions and theorems on time scales can be found in [15] which are excellent references for the calculus of time scales.

In recent years, there is much attention paid to the existence of positive solution for second-order multipoint and higher-order two-point boundary value problems on time scales; for details, see [616] and references therein. However, to the best of our knowledge, there are not many results concerning multipoint boundary value problems of higher-order on time scales; we refer the readers to [1720] for some recent results.

We would like to mention some results of Anderson and Avery [17], Anderson and Karaca [18], Han and Liu [19], and Yaslan [20]. In [17], Anderson and Avery studied the following even-order three-point BVP:

They have studied the existence of at least one positive solution to the BVP (1.4) using the functional-type cone expansion-compression fixed point theorem.

In [18], Anderson and Karaca were concerned with the dynamic three-point boundary value problem (1.2) and the eigenvalue problem with the same boundary conditions where is a positive parameter. Existence results of bounded solutions of a noneigenvalue problem were first established as a result of the Schauder fixed point theorem. Second, the monotone method was discussed to ensure the existence of solutions of the BVP (1.2). Third, they established criteria for the existence of at least one positive solution of the eigenvalue problem by using the Krasnosel’skii fixed point theorem. Later, they investigated the existence of at least two positive solutions of the BVP (1.2) by using the Avery-Henderson fixed point theorem.

In [19], Han and Liu studied the existence and uniqueness of nontrivial solution for the following third-order -Laplacian -point eigenvalue problems on time scales:

where is -Laplacian operator, that is, is a parameter, and . They obtained several sufficient conditions of the existence and uniqueness of nontrivial solution of the BVP (1.5) when is in some interval. Their approach was based on the Leray-Schauder nonlinear alternative.

Very recently, Yaslan [20] investigated the existence of solutions to the nonlinear even-order three-point boundary value problem on time scales :

for , where and are given constants. On the one hand, the author established criteria for the existence of at least one solution and of at least one positive solution for the BVP (1.6) by using the Schauder fixed point theorem and Krasnosel’skii fixed point theorem, respectively. On the other hand, the author investigated the existence of multiple positive solutions to the BVP (1.6) by using Avery-Henderson fixed point theorem and Leggett-Williams fixed point theorem.

In this paper, motivated by [21], firstly, a new existence result for (1.1) is obtained by using a fixed point theorem, which is due to and [22]. Particularly, may not be sublinear. Secondly, some simple criteria for the existence of a nonnegative solution of the BVP (1.2) are established by using Leray-Schauder nonlinear alternative. Thirdly, we investigate the existence of a nontrivial solution of the BVP (1.2); our approach is also based on the application of Leray-Schauder nonlinear alternative. Particularly, we do not require any monotonicity and nonnegativity on . Our conditions imposed on are all very easy to verify; our method is motivated by [1, 21, 23, 24].

2. Preliminaries

To state and prove the main results of this paper, we need the following lemmas.

Lemma 2.1 (see [18]). For , let be Green's function for the following boundary value problem: and let . Then, for , where

Lemma 2.2 (see [18]). Under condition , for , Green’s function in (2.2) possesses the following property:

Lemma 2.3 (see [18]). Assume that holds. Then, for , Green’s function in (2.2) satisfies

Lemma 2.4 (see [18]). Assume that condition is satisfied. For as in (2.2), take and recursively define for . Then is Green’s function for the homogeneous problem:

Lemma 2.5 (see [18]). Assume that holds. If one defines , then the Green function in Lemma 2.4 satisfies the following inequalities: where

Lemma 2.6 (see [22]). Let be a Banach space and let be completely continuous. If there exists a bounded and linear operator such that is not an eigenvalue of and then has a fixed point in .

Lemma 2.7 (see [25]). Let be a real Banach space, let be a bounded open subset of , , and let be a completely continuous operator. Then either there exist , such that or there exists a fixed point .

Suppose that denotes the Banach space with the norm .

3. Existence Results

In this section, we apply Lemmas 2.6 and 2.7 to establish some existence criteria for (1.1) and (1.2).

Theorem 3.1. Suppose that condition holds, and is continuous with . If then the BVP (1.1) has a solution , and when .

Proof. Define the integral operator by for . Obviously, the solutions of the BVP (1.1) are the fixed points of operator . From the proof of Theorem of [18], we can know that is completely continuous.
In order to apply Lemma 2.6, we consider the following BVP:
Define the integral operator by for . Then it is easy to check that is completely continuous (so bounded) linear operator and that solutions of the BVP (3.3) are the fixed points of operator and conversely.
First, we claim that 1 is not an eigenvalue of .
In fact, if , then it is obvious that the BVP (3.3) has no nontrivial solution.
If and the BVP (3.3) has a nontrivial solution , then , and so
which is impossible. So, 1 is not an eigenvalue of .
Next, we will show that
In fact, for any , since , there must exist a number such that Let Then for any and (), we distinguish the following two cases.Case 1 (). In this case, choose such that Thus, when and , we have which together with (3.7) implies that Case 2 (). In this case, when , from (3.7), we see that Thus, we can deduce from (3.11) and (3.12) that for any and From (3.13), we have that is to say, Then, it follows from Lemma 2.6 that has a fixed point . In other words, is a solution of the BVP (1.1). Moreover, we can assert that is nontrivial when . In fact, if , then that is, 0 is not a solution of the BVP (1.1).

Corollary 3.2. Assume that condition holds, and is continuous with . Then the BVP (1.1) has a nonnegative solution.

Proof. Let then is continuous, and from , we know that Consider the following BVP: It follows from Theorem 3.1 that the BVP (3.19) has a solution , that is, Since and are nonnegative, we can get that on . Consequently, from the definition of , we have It follows from the boundary conditions of (3.20) and (3.21) that is a nonnegative solution of the BVP (1.1).

Remark 3.3. In Corollary 3.2, we only need that . Thus, may not be sublinear.

Theorem 3.4. Assume that condition holds, and where and are defined in Lemmas 2.5 and 2.1, respectively. Then the BVP (1.2) has a nonnegative solution with.

Proof. We consider the following boundary value problem: where , and Let be any solution of (3.25). Then for . We note that for . It follows from condition (3.23) in Theorem 3.4 that for Consequently, which together with the condition (3.24) in Theorem 3.4 implies that .
Let be given by
It is easy to show that is completely continuous.
Let
Since , any solution of with cannot occur. Lemma 2.7 guarantees that has a fixed point in . In other words, the BVP (1.2) has a solution with .

Theorem 3.5. Assume that condition is satisfied. Suppose that , , is continuous, and there exist nonnegative integrable functions , such that Then, the BVP (1.2) has at least one nontrivial solution .

Proof. Let By hypothesis . Since , there exists such that . On the other hand, from the condition , we know that .
Let , . For , the operator is defined by
from the proof of Theorems 3.1 and 3.4, we have known that is a completely continuous operator, and the BVP (1.2) has at least one nontrivial solution if and only if is a fixed point of in .
Suppose , such that then
Therefore which contradicts . By Lemma 2.7, has a fixed point . Noting , the BVP (1.2) has at least one nontrivial solution . This completes the proof.

Corollary 3.6. Assume that condition is satisfied. Suppose that , , is continuous, and there exist nonnegative integrable functions such that Then, the BVP (1.2) has at least one nontrivial solution .

Proof. In this case, we have By Theorem 3.5, this completes the proof.

Corollary 3.7. Assume that condition is satisfied. Suppose that , , is continuous, and there exist nonnegative integrable functions k, h such that Then, the BVP (1.2) has at least one nontrivial solution .

Proof. Let then, there exists such that Set and it follows from (3.41) and (3.42) that By Corollary 3.6, we can deduce that Corollary 3.7 is true.

4. Two Examples

In the section, we present two examples to explain our results.

Example 4.1. Let , and consider the following BVP: where It is easy to check that and therefore, the condition is satisfied. By computation, we can get that where From the proof of Lemma in [18], we can get that and for and , we have Thus, we can obtain that By Theorem 3.1, it is easy to get that the BVP (4.1) has a solution .

Example 4.2. Let us introduce an example to illustrate the usage of Theorem 3.5. Let and . Then condition is satisfied. Green’s function in Lemma 2.1 is where Green’s function in Lemma 2.1 is where Since , by using the cases in the proof of Lemma of [18], we can know that . Therefore, we have .
Set and . Then it is easy to prove that
On the other hand, since , we can know that by using the cases in the proof of Lemma of [18]. Therefore, we have Thus Hence, by Theorem 3.5, the BVP (1.2) has at least one nontrivial solution .

Acknowledgment

Project supported by the Youth Science Foundation of Shanxi Province (2009021001-2).

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