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.

In [3], Anderson and Karaca discussed the dynamic equation on time scales

(1.1)

and the eigenvalue problem

(1.2)

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.

In [4], by using the Gatica-Oliker-Waltman fixed-point theorem, DaCunha, Davis, and Singh proved the existence of a positive solution for the three-point boundary value problem on a time scale given by

(1.3)

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

Anderson et al. [5] gave a detailed presentation for the following higher-order self-adjoint boundary value problem on time scales:

(1.4)

and got many excellent results.

In related papers, Sun [11] considered the following third-order two-point boundary value problem on time scales:

(1.5)

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

In this paper, we consider the existence of positive solutions for the following higher-order four-point singular boundary value problem (BVP) on time scales

(1.6)
(1.7)

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

For convenience, we list the following definitions which can be found in [1, 2, 9, 14, 17]. A time scale is a nonempty closed subset of real numbers . For and , define the forward jump operator and backward jump operator , respectively, by

(2.1)

for all . If , is said to be right scattered, and if , is said to be left scattered; if , is said to be right dense, and 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 . In this general time-scale setting, represents the delta (or Hilger) derivative [13, Definition 1.10],

(2.2)

where is the forward jump operator, is the forward graininess function, and is abbreviated as . In particular, if , then and , while if for any , then and

(2.3)

A function is right-dense continuous provided that it is continuous at each right-dense point (a point where ) and has a left-sided limit at each left-dense point . The set of right-dense continuous functions on is denoted by . It can be shown that any right-dense continuous function has an antiderivative (a function with the property for all ). Then the Cauchy delta integral of is defined by

(2.4)

where is an antiderivative of on . For example, if , then

(2.5)

and if , then

(2.6)

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.

If and then

(2.7)

In this paper, let

(2.8)

Then is a Banach space with the norm . Define a cone by

(2.9)

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

Lemma 2.2.

Suppose condition holds. Then there exists a constant satisfies

(2.10)

Furthermore, the function

(2.11)

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

Lemma 2.3.

Let and in Lemma 2.2. Then

(2.12)

Proof.

Suppose .

We will discuss it from three perspectives.

(i). It follows from the concavity of that

(2.13)

then

(2.14)

which means .

(ii). If , we have

(2.15)

then

(2.16)

If , we have

(2.17)

then

(2.18)

and this means .

(iii). Similarly, we have

(2.19)

then

(2.20)

which means .

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

Lemma 2.4.

Suppose that conditions hold, then is a solution of boundary value problem (1.6), (1.7) if and only if is a solution of the following integral equation:

(2.21)

where

(2.22)

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

(2.23)

thus

(2.24)

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

(2.25)

By the equation of the boundary condition (1.7), we get

(2.26)

then

(2.27)

Secondly, by (2.24) and let on (2.24), we have

(2.28)

Then

(2.29)

Then by delta integrating (2.29) for times on , we have

(2.30)

Similarly, for , by delta integrating the equation of problems (1.6) on , we have

(2.31)

Therefore, for any , can be expressed as the equation

(2.32)

where is expressed as (2.22).

Sufficiency. Suppose that

(2.33)

then by (2.22), we have

(2.34)

So,

(2.35)

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.

Now, we define a mapping given by

(2.36)

where is given by (2.22).

Lemma 2.5.

Suppose that conditions hold, the solution of problem (1.6), (1.7) satisfies

(2.37)

and for in Lemma 2.2, one has

(2.38)

Proof.

If is the solution of (1.6), (1.7), then is a concave function, and , thus we have

(2.39)

that is,

(2.40)

By Lemma 2.3, for , we have

(2.41)

then . The proof is complete.

Lemma 2.6.

is completely continuous.

Proof.

Because

(2.42)

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

(2.43)

where is the constant from Lemma 2.2. By Lemma 2.5, we can also set

(2.44)

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.

Without loss of generality, we suppose that . For any , by Lemma 2.3, we have

(3.1)

We define two open subsets and of :

(3.2)

For any , by (3.1) we have

(3.3)

For and , we will discuss it from three perspectives.

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

(3.4)
  1. (ii)

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

(3.5)
  1. (iii)

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

(3.6)

Therefore, no matter under which condition, we all have

(3.7)

Then by Theorem 2.1, we have

(3.8)

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

(3.9)

thus

(3.10)

Then, by Theorem 2.1, we get

(3.11)

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

(3.12)

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

Proof of Theorem 3.2.

First, by , for , there exists an adequately small positive number , as , we have

(3.13)

Then let , thus by (3.13)

(3.14)

So condition () holds. Next, by condition (), , then for , there exists an appropriately big positive number , as , we have

(3.15)

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.

Firstly, by condition (), , then for , there exists an adequately small positive number , as , we have

(3.16)

thus when , we have

(3.17)

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

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

(3.18)

If is unbounded, by the continuity of on , then there exist a constant , and a point such that

(3.19)

Thus, by we know

(3.20)

Choose . Then, we have

(3.21)

If is bounded, we suppose , there exists an appropriately big positive number , then choose , we have

(3.22)

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.

Consider the following boundary value problem on the specific time scale:

(4.1)

where

(4.2)

and is the constant defined in Lemma 2.2,

(4.3)

Then obviously

(4.4)

By Theorem 2.1, we have

(4.5)

so conditions (), hold.

By simple calculations, we have

(4.6)

then , that is, , so condition holds.

For , it is easy to see that

(4.7)

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

Example 4.2.

Consider the following boundary value problem on the specific time scale .

(4.8)

where

(4.9)

and is the constant from Lemma 2.2,

(4.10)

Then obviously

(4.11)

By Theorem 2.1, we have

(4.12)

so conditions (), hold. By simple calculations, we have

(4.13)

then , that is, , so condition holds.

For , it is easy to see that

(4.14)

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