- Research Article
- Open Access

# Existence Results for Higher-Order Boundary Value Problems on Time Scales

- Jian Liu
^{1}Email author and - Yanbin Sang
^{2}

**2009**:209707

https://doi.org/10.1155/2009/209707

© J. Liu and Y. Sang. 2009

**Received:**22 March 2009**Accepted:**16 June 2009**Published:**29 July 2009

## Abstract

By using the fixed-point index theorem, we consider the existence of positive solutions for the following nonlinear higher-order four-point singular boundary value problem on time scales , ; , ; , ; , , where , , , , , , , and is rd-continuous.

## Keywords

- Boundary Value Problem
- Small Positive Number
- Singular Boundary
- Jump Operator
- Nonempty Closed Subset

## 1. Introduction

Time scales and time-scale notationare introduced well in the fundamental texts by Bohner and Peterson [1, 2], respectively, as important corollaries. In, the recent years, many authors have paid much attention to the study of boundary value problems on time scales (see, e.g., [3–17]). In particular, we would like to mention some results of Anderson et al. [3, 5, 6, 14, 16], DaCunha et al. [4], and Agarwal and O'Regan [7], which motivate us to consider our problem.

with the same boundary conditions where is a positive parameter. They obtained some results for the existence of positive solutions by using the Krasnoselskii, the Schauder, and the Avery-Henderson fixed-point theorem.

where is fixed, and is singular at and possibly at .

and got many excellent results.

where and . Some existence criteria of solution and positive solution are established by using the Leray-Schauder fixed point theorem.

where , , and is rd-continuous. In the rest of the paper, we make the following assumptions:

()

() .

In this paper, by constructing one integral equation which is equivalent to the BVP (1.6) and (1.7), we study the existence of positive solutions. Our main tool of this paper is the following fixed-point index theorem.

Theorem 1.1 ([18]).

Suppose is a real Banach space, is a cone, let . Let operator be completely continuous and satisfy . Then

(i)if , then

(ii)if , then .

The outline of the paper is as follows. In Section 2, for the convenience of the reader we give some definitions and theorems which can be found in the references, and we present some lemmas in order to prove our main results. Section 3 is developed in order to present and prove our main results. In Section 4 we present some examples to illustrate our results.

## 2. Preliminaries and Lemmas

Throughout we assume that are points in , and define the time-scale interval . In this paper, we also need the the following theorem which can be found in [1].

Theorem 2.1.

Obviously, is a cone in . Set . If on then we say is concave on We can get the following.

Lemma 2.2.

is a positive continuous function on , therefore has minimum on . Then there exists such that .

Lemma 2.3.

Proof.

Suppose .

We will discuss it from three perspectives.

(i) . It follows from the concavity of that

which means .

(ii) . If , we have

If , we have

and this means .

(iii) . Similarly, we have

which means .

From the above, we know . The proof is complete.

Lemma 2.4.

Proof.

*Necessity.*By the equation of the boundary condition, we see that , then there exists a constant such that . Firstly, by delta integrating the equation of the problems (1.6) on , we have

By and the boundary condition (1.7), let on (2.23), we have

where is expressed as (2.22).

*Sufficiency*. Suppose that

which imply that (1.6) holds. Furthermore, by letting and on (2.22) and (2.34), we can obtain the boundary value equations of (1.7). The proof is complete.

where is given by (2.22).

Lemma 2.5.

Proof.

By Lemma 2.3, for , we have

then . The proof is complete.

Lemma 2.6.

is completely continuous.

Proof.

is continuous, decreasing on and satisfies . Then, for each and . This shows that . Furthermore, it is easy to check that is completely continuous by Arzela-ascoli Theorem.

For convenience, we set

## 3. The Existence of Positive Solution

Theorem 3.1.

Suppose that conditions ( ), ( ) hold. Assume that also satisfies

where

Then, the boundary value problem (1.6), (1.7) has a solution such that lies between and .

Theorem 3.2.

Suppose that conditions ( ), ( ) hold. Assume that also satisfies

Then, the boundary value problem (1.6), (1.7) has a solution such that lies between and .

Theorem 3.3.

Suppose that conditions ( ), ( ) hold. Assume that also satisfies

()

()

Then, the boundary value problem (1.6), (1.7) has a solution such that lies between and .

Proof of Theorem 3.1.

For and , we will discuss it from three perspectives.

(i)If , thus for , by ( ) and Lemma 2.4, we have

- (ii)
If , thus for , by ( ) and Lemma 2.4, we have

- (iii)
If , thus for , by ( ) and Lemma 2.4, we have

On the other hand, for , we have ; by ( ) we know

Therefore, by (3.8), (3.11), , we have

Then operator has a fixed point , and . Then the proof of Theorem 3.1 is complete .

Proof of Theorem 3.2.

Let , thus by (3.15), condition ( ) holds. Therefore by Theorem 3.1 we know that the results of Theorem 3.2 hold. The proof of Theorem 3.2 is complete.

Proof of Theorem 3.3.

Let , so by (3.17), condition ( ) holds.

Secondly, by condition ( ), , then for , there exists a suitably big positive number , as , we have

Therefore, condition ( ) holds. Thus, by Theorem 3.1, we know that the result of Theorem 3.3 holds. The proof of Theorem 3.3 is complete.

## 4. Application

In this section, in order to illustrate our results, we consider the following examples.

Example 4.1.

By Theorem 2.1, we have

so conditions ( ), hold.

By simple calculations, we have

then , that is, , so condition holds.

For , it is easy to see that

so condition holds. Then by Theorem 3.2, BVP (4.1) has at least one positive solution.

Example 4.2.

then , that is, , so condition holds.

For , it is easy to see that

then condition holds. Thus by Theorem 3.3, BVP (4.8) has at least one positive solution.

## Declarations

### Acknowledgment

The authors would like to thank the anonymous referee for his/her valuable suggestions, which have greatly improved this paper.

## Authors’ Affiliations

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