Journal of Applied MathematicsVolume 2012 (2012), Article ID 939162, 19 pagesdoi:10.1155/2012/939162

Research Article

Higher-Order Dynamic Delay Differential Equations on Time Scales

Hua Su,¹ Lishan Liu,² and Xinjun Wang³

¹School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China²School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China³School of Economics, Shandong University, Jinan 250014, China

Received 11 October 2011; Accepted 12 February 2012

Academic Editor: Yansheng Liu

Copyright © 2012 Hua Su et al. 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 study the existence of positive solutions for the nonlinear four-point singular boundary value problem with higher-order -Laplacian dynamic delay differential equations on time scales, subject to some boundary conditions. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with -Laplacian operator are obtained.

1. Introduction

The study of dynamic equations on time scales goes back to its founder Stefan Hilger [1] and is a new area of still fairly theoretical exploration in mathematics. Boundary value problems for delay differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory (see [2, 3]). In recent years, many authors have begun to pay attention to the study of boundary value problems or with -Laplacian equations or with -Laplacian dynamic equations on time scales (see [4–19] and the references therein).

In [7], Sun and Li considered the existence of positive solution of the following dynamic equations on time scales: where , , , . They obtained the existence of single and multiple positive solutions of the problem (1.1) by using fixed point theorem and Leggett-Williams fixed point theorem, respectively.

In [10], Avery and Anderson discussed the following dynamic equation on time scales: He obtained some results for the existence of one positive solution of the problem (1.2) based on the limits and .

In [11], Wang et al. discussed the following dynamic equation by using Avery-Peterson fixed theorem (see [10]): They obtained some results for the existence, three positive solutions of the problem (1.3), (1.4) and (1.3), (1.5), respectively.

However, there are not many concerning the -Laplacian problems on time scales. Especially, for the singular multi point boundary value problems for higher-order -Laplacian dynamic delay differential equations on time scales, with the author’s acknowledg, no one has studied the existence of positive solutions in this case.

Recently, in [16], we study the existence of positive solutions for the following nonlinear two-point singular boundary value problem with -Laplacian operator by using the fixed point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for nonlinear singular boundary value problem (1.6) with -Laplacian operator are obtained.

Now, motivated by the results mentioned above, in this paper, we study the existence of positive solutions for the following nonlinear four-point singular boundary value problem with higher-order -Laplacian dynamic delay differential equations operator on time scales (SBVP): where is -Laplacian operator, that is, , , , ; , is prescribed and , , , , , .

In this paper, by constructing one integral equation which is equivalent to the problem (1.7), (1.8), we research the existence of positive solutions for nonlinear singular boundary value problem (1.7), (1.8) when and satisfy some suitable conditions.

Our main tool of this paper is the following fixed point index theory.

Theorem 1.1 (see [18]). Suppose is a real Banach space, is a cone, let . Let operator be completely continuous and satisfy , for all . Then (i)if , for all , then ; (ii)if , for all , then .

This paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, we study the existence of at least two solutions of the systems (1.7), (1.8). In Section 4, we give an examples as the application.

2. Preliminaries and Lemmas

A time scale is an arbitrary nonempty closed subset of real numbers . In [1, 14, 20], we can find some basic definitions about time scale. The operators and from to : are called the forward jump operator and the backward jump operator, respectively.

If , then . If , then is the forward difference operator, while is the backward difference operator.

A function is left-dense continuous (i.e., -continuous), if is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in . It is well known that is -continuous.

If , then we define the nabla integral by If , then we define the delta integral by

In the rest of this paper, is closed subset of with , . And let

Here, Then is a Banach space with the norm . And let

Obviously, is a cone in . Set .

Definition 2.1. is called a solution of SBVP (1.7) and (1.8) if it satisfies the following:(1);(2) for all and satisfy conditions (1.8);(3) hold for .In the rest of the paper, we also make the following assumptions:();() and there exists , such that(), on and .It is easy to check that condition () implies that We can easily get the following lemmas.

Lemma 2.2. Suppose condition holds. Then there exists a constant satisfing Furthermore, the function is positive continuous functions on ; therefore, has minimum on . Hence we suppose, that there exists such that , .

Proof. At first, it is easily seen that is continuous on . Next, let Then, from condition , we have that the function is strictly monotone nondecreasing on and , the function is strictly monotone nonincreasing on and , which implies . The proof is complete.

Lemma 2.3. Let and of Lemma 2.2, then The proof of the above lemma is similar to the proof in [17, Lemma 2.2], so we omit it.

Lemma 2.4. Suppose that conditions hold, is a solution of the following boundary value problems: where Then, , is a positive solution to the SBVP (1.7) and (1.8).

Proof. It is easy to check that satisfies (1.7) and (1.8).

So in the rest of the sections of this paper, we focus on SBVP (2.13) and (2.14).

Lemma 2.5. Suppose that conditions hold, is a solution of boundary value problems (2.13), (2.14) if and only if is a solution of the following integral equation: where

Proof. Necessity. Obviously, for , we have . If , by the equation of the boundary condition, we have , , then there exists a constant such that .Firstly, by integrating the equation of the problems (2.13) on , we have then thus By and condition (2.18), let on (2.18), we have By the equation of the boundary condition (2.14), we have then Then, by (2.20) and leting on (2.20), we have Then Then, by integrating (2.25) for times on , we have Similarly, for , by integrating the equation of problems (2.13) on , we have Therefore, for any , can be expressed as equation where is expressed as (2.17). Then the results of Lemma 2.3 hold.Sufficiency. Suppose that , . Then by (2.17), we have So, , . These imply that (2.13) holds. Furthermore, by letting and on (2.17) and (2.29), we can obtain the boundary value equations of (2.14). The proof is complete.

Now, we define an operator equation given by where is given by (2.17).

From the definition of and the previous discussion, we deduce that, for each , . Moreover, we have the following lemmas.

Lemma 2.6. is completely continuous.

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

Lemma 2.7. Suppose that conditions hold, the solution of problem (2.13), (2.14) satisfies and for in Lemma 2.2, one has

Proof. Firstly, we can have Next, if is the solution of problem (2.13), (2.14), then is concave function, and , . Thus, we have that is, , .Finally, by Lemma 2.3, for , we have . By , we have The proof is complete.

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

3. The Existence of Multiple Positive Solutions

In this section, we also make the following conditions:(), for , ;(), for , .

Next, we will discuss the existence of multiple positive solutions.

Theorem 3.1. Suppose that conditions (), (), (), and () hold. Assume that also satisfies ();().Then, the SBVP (2.13), (2.14) hase at last two solutions , such that

Proof. For any , by Lemma 2.3, we have First, by condition (), for any , there exists a constant such that Set . For any , by (3.2) we have For any , by (3.3) and Lemmas 2.3–2.6, we will discuss it from three perspectives.(i)If , we have (ii)If , we have(iii)If , we have Therefore, no matter under which condition, we all have Then, by Theorem 1.1, we have Next, by condition (), for any , there exists a constant such that We choose a constant , obviously . Set . For any , by Lemma 2.3, we have Then, by (3.10), Lemmas 2.3–2.6 and also similar to the previous proof, we can also have from three perspectives that Then, by Theorem 1.1, we have Finally, set . For any , we have , by () we know Thus, Then, by Theorem 1.1, we have Therefore, by (3.9), (3.13), (3.16), we have Then has fixed point and fixed point . Obviously, are all positive solutions of problem (2.13), (2.14) and . Proof of Theorem 3.1 is complete.

Theorem 3.2. Suppose that conditions (), (), (), () hold. Assume that also satisfies ();().Then, the SBVP (2.13), (2.14) has at last two solutions such that .

Proof. First, by , for , there exists a constant such that Set , for any , by (3.18), we have that is, Then, by Theorem 1.1, we have Next, let ; note that is monotone increasing with respect to . Then, from , it is easy to see that Therefore, for any , there exists a constant such that Set , for any , by (3.23), we have that is, Then, by Theorem 1.1, we have Finally, set . For any , by , Lemma 2.3 and also similar to the previous proof of Theorem 3.1, we can also have Then, by Theorem 1.1, we have Therefore, by (3.21), (3.28), (3.26), , we have Then has fixed point and fixed point . Obviously, are all positive solutions of problem (2.13), (2.14) and . The proof of Theorem 3.2 is complete.

Similar to Theorems 3.1 and 3.2, we also obtain the following theorems.

Theorem 3.3. Suppose that conditions (), (), (), and () hold and (),().Then, the SBVP (2.13), (2.14) has at last two solutions such that .

Theorem 3.4. Suppose that conditions (), (), (), and () hold and ();().Then, the SBVP (2.13), (2.14) has at last two solutions such that .

4. An Example

Example 4.1. Consider the following 3-order singular boundary value problem (SBVP) with -Laplacian: where So, by Lemma 2.4, we discuss the following SBVP: where Then, obviously, so conditions (), , , (), and hold.Next, we choose , and for , because of the monotone increasing of on , then Therefore, by we know so condition holds. Then, by Theorem 3.1, SBVP (4.3) has at least two positive solutions and . Then, by Lemma 2.4, , , are the positive solutions of the SBVP (4.1).

Acknowldgments

The first and second authors were supported financially by Shandong Province Natural Science Foundation (ZR2009AQ004), and National Natural Science Foundation of China (11071141), and the third author was supported by Shandong Province planning Foundation of Social Science (09BJGJ14).

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