Variational Inequalities and Vector Optimization 2014View this Special Issue
Research Article | Open Access
Solutions of Second-Order -Point Boundary Value Problems for Impulsive Dynamic Equations on Time Scales
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.
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  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  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  and Krasnosel’skii fixed point theorem . 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.
In this section we state the preliminary information that we need to prove the main results. From Lemmas 2.1 and 2.2 in , 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 .
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 ). 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 9 (see ). 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
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.