Journal of Applied Mathematics

Volume 2014 (2014), Article ID 867018, 10 pages

http://dx.doi.org/10.1155/2014/867018

Research Article

Solutions of Second-Order -Point Boundary Value Problems for Impulsive Dynamic Equations on Time Scales

Xue Xu1,2 and Yong Wang¹

¹Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

²Department of Mathematics, Harbin University, Harbin, Heilongjiang 150086, China

Received 24 December 2013; Accepted 16 January 2014; Published 10 April 2014

Academic Editor: Xian-Jun Long

Copyright © 2014 Xue Xu and Yong Wang. 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 a general second-order -point boundary value problems for nonlinear singular impulsive dynamic equations on time scales , , and The existence and uniqueness of positive solutions are established by using the mixed monotone fixed point theorem on cone and Krasnosel’skii fixed point theorem. In this paper, the function items may be singular in its dependent variable. We present examples to illustrate our results.

1. Introduction

The theory of dynamic equations on time scales unifies the well-known analogies in the concept of difference equations and differential equations. In the past few years, the boundary value problems of dynamic equations on time scales have been studied by many authors (see [1–16] and references cited therein). Some classical tools have been used in the literature to study dynamic equations. These classical tools include the coincidence degree theory [11, 12], the method of upper and lower solutions [7, 10], and some fixed point theorems in cones for completely continuous operators [1–5, 9, 13–16]. Recently, multiple-point boundary value problems on time scale have been studied for instance [4, 5, 12].

In 2008, Lin and Du [5] studied the -point boundary value problem for second-order dynamic equations on time scales:

where is a time scale. By using Green’s function and the Leggett-Williams fixed point theorem in an appropriate cone, the existence of at least three positive solutions of the problem is obtained.

In 2009, Topal and Yantir [4] studied the general second-order nonlinear -point boundary value problems:

with no singularity. The authors deal with the determining the value of , the existence of multiple positive solutions of (2) is obtained by using the Krasnosel’skii and Legget-William fixed point theorems.

Impulsive differential equations are now recognized as an excellent source of models for simulating processes and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnology, industrial robotics, optimal control, and so forth. In recent years, impulsive differential equations have become a very active area of research. In this paper, we consider the following impulsive singular dynamic equations on time scales:

where , for all , , and satisfy the following:

(C1) and may be singular at , ;(C2) and there exists such that , may be singular at ;(C3), .

The main theorems of this paper complement the very little existence results devoted to impulsive dynamic equations on a time scale. We will prove our two existence results for the problem (3) by using mixed monotone fixed point theorem on cone [17] and Krasnosel’skii fixed point theorem [18]. This paper is organized as follows. In Section 2, starting with some preliminary lemmas, we state a mixed monotone fixed point theorem on cone and Krasnosel’skii fixed point theorem. In Section 3, we give the main result which states the sufficient conditions for the -point boundary value problem (3) to have existence of positive solutions. We also present examples to illustrate our results work.

2. Preliminaries

In this section we state the preliminary information that we need to prove the main results. From Lemmas 2.1 and 2.2 in [4], we have the following lemma.

Lemma 1. Assume that (C3) holds. Then the equations

have unique solutions and , respectively, and

(a)    is strictly increasing on ; (b) is strictly decreasing on .

For the rest of the paper we need the following assumption:

(C4).

Lemma 2. Assume that (C3) and (C4) hold. Let . Then boundary value problem

is equivalent to the integral equation

where

Lemma 3. Green’s function has the properties

Lemma 4. Assume that (C3) and (C4) hold. Let be a solution of the boundary value problem (3) if and only if is a solution of the following impulsive integral equation:

where

Lemma 5. Green’s function defined by (10) has the properties

where

Lemma 6. Assume that (C1)–(C4) hold. Then the solution of (3) satisfies .

The proofs of the Lemmas 3–6 can be obtained easily by Lemmas 1 and 2.

For our constructions, we will consider the Banach space equipped with standard norm . We define a cone by

From Lemmas 4 and 5, we define the integral operator by

Then, it is clear that the solutions of (3) are the fixed points of the operator .

Thus, from Lemma 4, standard arguments show that and is completely continuous.

The following content will play major role in our next analysis.

Let be a normal cone of a Banach space , and let with . Define

Definition 7 (see [17]). Assume . is said to be mixed monotone if is nondecreasing in and nonincreasing in , that is, if implies for any and implies for any . is said to be a fixed point of if .

Theorem 8 (see [17, 19]). Suppose that is a mixed monotone operator and a constant , such that

Then has a unique fixed point . Moreover, for any ,

satisfy

where

, and is a constant from .

Theorem 9 (see [18]). Let be a Banach space, and let be a cone in . Assume that are open subsets of with , and let be a completely continuous operator such that either

(1), , , , or(2), , .

Then has a fixed point in .

3. Main Results

First, by using Theorem 8 we establish the following main result.

Theorem 10. Suppose that conditions hold and

(C5), () and

 for .(C6)There exists such that

 for any and, , .(C7)Consider that satisfies

Then (3) has an unique positive solution .

Proof. Since (21) holds, let ; one has

Then

Let . The above inequality is

From (21), (25), and (26), one has

Similarly, from (22), one has

Let ; one has

Let ; it is clear that . We define

where is chosen such that

For any , we define

Let . On one hand, from (27), (28), and (29), for , we have

then

Thus, for any , we have

So, is well defined and .

Next, for any , one has

So the conditions of Theorems 8 hold. Therefore there exists a unique such that . This completes the proof of Theorem 10.

Example 11. Consider the following singular boundary value problem:

where ,, , and and satisfy , .

Let , , , and . Thus

Applying Theorem 10, we can find that the above equation has a unique solution .

In the next, using Theorem 9 we establish the following main result.

Theorem 12. Suppose that conditions hold and the following conditions are satisfied:

Here is Green’s function and

Then (3) has two nonnegative solutions with and for .

Proof. First we will show that there exists a solution to (3) with for and . Let

We now show that

To see this, let . Then and for . So for we have

Then

This together with (42) yields

so (47) is satisfied.

Next we show

To see this let so and for .

We have

Then

This together with (44) yields

Thus , so (51) holds.

Now Theorem 9 implies that has a fixed point ; that is, and for . It follows from (47) and (51) that , so we have .

Similarly, if we put

we can show that there exists a solution to (3) with for and . This completes the proof of Theorem 12.

Similar to the proof of Theorem 12, we have the following result.

Theorem 13. Suppose that conditions and (39)–(43) hold. In addition suppose that

Then (3) has a nonnegative solution with and for .

Remark 14. If in (56) we have , then (3) has a nonnegative solution with and for .

It is easy to use Theorem 12 and Remark 14 to write theorems which guarantee the existence of more than two solutions to (3). We state one such result.

Theorem 15. Suppose that conditions (C1)–(C4); (39)–(41); and (43) hold. Assume that and constants , with , and

In addition suppose for each that

hold. Then (3) has nonnegative solutions with for .

Example 16. Consider the boundary value problem

where   , and is such that

and here

Then (59) has two solutions with for .

To see this we will apply Theorem 12 with (here will be chosen below)

Clearly (39)–(41) and (43) hold, and

Now (42) holds with since

Finally, let

and notice that (44) is satisfied for small and large since

as , since . Thus all the conditions of Theorem 12 are satisfied so existence is guaranteed.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgment

The work was supported by Youth academic backbone project of Heilongjiang Provincial University (no. 1252G035).

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