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

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## 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 Open image in new window , Open image in new window ; Open image in new window , Open image in new window ; Open image in new window , Open image in new window ; Open image in new window , Open image in new window , where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window 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, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 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 Open image in new window 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 Open image in new window is fixed, and Open image in new window is singular at Open image in new window and possibly at Open image in new window .

and got many excellent results.

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

where Open image in new window , Open image in new window Open image in new window , and Open image in new window 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 Open image in new window is a real Banach space, Open image in new window is a cone, let Open image in new window . Let operator Open image in new window Open image in new window be completely continuous and satisfy Open image in new window . Then

(i)if Open image in new window , then Open image in new window

(ii)if Open image in new window , then Open image in new window .

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 Open image in new window are points in Open image in new window , and define the time-scale interval Open image in new window . In this paper, we also need the the following theorem which can be found in [1].

Theorem 2.1.

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

Lemma 2.2.

is a positive continuous function on Open image in new window , therefore Open image in new window has minimum on Open image in new window . Then there exists Open image in new window such that Open image in new window .

Lemma 2.3.

Proof.

Suppose Open image in new window .

We will discuss it from three perspectives.

(i) Open image in new window . It follows from the concavity of Open image in new window that

which means Open image in new window .

(ii) Open image in new window . If Open image in new window , we have

If Open image in new window , we have

and this means Open image in new window .

(iii) Open image in new window . Similarly, we have

which means Open image in new window .

From the above, we know Open image in new window . The proof is complete.

Lemma 2.4.

Proof.

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

By Open image in new window and the boundary condition (1.7), let Open image in new window on (2.23), we have

where Open image in new window is expressed as (2.22).

*Sufficiency*. Suppose that

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

where Open image in new window is given by (2.22).

Lemma 2.5.

Proof.

By Lemma 2.3, for Open image in new window , we have

then Open image in new window . The proof is complete.

Lemma 2.6.

Open image in new window is completely continuous.

Proof.

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

For convenience, we set

## 3. The Existence of Positive Solution

Theorem 3.1.

Suppose that conditions ( Open image in new window ), ( Open image in new window ) hold. Assume that Open image in new window also satisfies

where Open image in new window

Then, the boundary value problem (1.6), (1.7) has a solution Open image in new window such that Open image in new window lies between Open image in new window and Open image in new window .

Theorem 3.2.

Suppose that conditions ( Open image in new window ), ( Open image in new window ) hold. Assume that Open image in new window also satisfies

Then, the boundary value problem (1.6), (1.7) has a solution Open image in new window such that Open image in new window lies between Open image in new window and Open image in new window .

Theorem 3.3.

Suppose that conditions ( Open image in new window ), ( Open image in new window ) hold. Assume that Open image in new window also satisfies

Then, the boundary value problem (1.6), (1.7) has a solution Open image in new window such that Open image in new window lies between Open image in new window and Open image in new window .

Proof of Theorem 3.1.

For Open image in new window and Open image in new window , we will discuss it from three perspectives.

(i)If Open image in new window , thus for Open image in new window , by ( Open image in new window ) and Lemma 2.4, we have

- (ii)
If Open image in new window , thus for Open image in new window , by ( Open image in new window ) and Lemma 2.4, we have

- (iii)
If Open image in new window , thus for Open image in new window , by ( Open image in new window ) and Lemma 2.4, we have

On the other hand, for Open image in new window , we have Open image in new window ; by ( Open image in new window ) we know

Therefore, by (3.8), (3.11), Open image in new window , we have

Then operator Open image in new window has a fixed point Open image in new window , and Open image in new window . Then the proof of Theorem 3.1 is complete .

Proof of Theorem 3.2.

Let Open image in new window , thus by (3.15), condition ( Open image in new window ) 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 Open image in new window , so by (3.17), condition ( Open image in new window ) holds.

Secondly, by condition ( Open image in new window ), Open image in new window , then for Open image in new window , there exists a suitably big positive number Open image in new window , as Open image in new window , we have

Therefore, condition ( Open image in new window ) 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 ( Open image in new window ), Open image in new window hold.

By simple calculations, we have

then Open image in new window , that is, Open image in new window , so condition Open image in new window holds.

For Open image in new window , it is easy to see that

so condition Open image in new window holds. Then by Theorem 3.2, BVP (4.1) has at least one positive solution.

Example 4.2.

then Open image in new window , that is, Open image in new window , so condition Open image in new window holds.

For Open image in new window , it is easy to see that

then condition Open image in new window holds. Thus by Theorem 3.3, BVP (4.8) has at least one positive solution.

## Notes

### Acknowledgment

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

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