Research Article | Open Access

# MultiPoint BVPs for Second-Order Functional Differential Equations with Impulses

**Academic Editor:**Fernando Lobo Pereira

#### Abstract

This paper is concerned about the existence of extreme solutions of multipoint boundary value problem for a class of second-order impulsive functional differential equations. We introduce a new concept of lower and upper solutions. Then, by using the method of upper and lower solutions introduced and monotone iterative technique, we obtain the existence results of extreme solutions.

#### 1. Introduction

In this paper, we consider the multipoint boundary value problems for the impulsive functional differential equation: where , . , is continuous everywhere except at ; and exist with ; . Denote . Let is continuous, and exist with ; is continuous differentiable, and exist with . Let . A function is called a solution of BVP(1.1) if it satisfies (1.1).

The method of upper and lower solutions combining monotone iterative technique offers an approach for obtaining approximate solutions of nonlinear differential equations [1–3]. There exist much literature devoted to the applications of this technique to general boundary value problems and periodic boundary value problems, for example, see [1, 4–6] for ordinary differential equations, [7–11] for functional differential equations, and [12] for differential equations with piecewise constant arguments. For the studies about some special boundary value problems, for example, Lidston boundary value problems and antiperiodic boundary value problems, one may see [13, 14] and the references cited therein.

Here, we hope to mention some papers where existence results of solutions of certain boundary value problems of impulsive differential equations were studied [11, 15] and certain multipoint boundary value problems also were studied [6, 16–21]. These works motivate that we study the multipoint boundary value problems for the impulsive functional differential equation (1.1).

We also note that when and , the boundary value problem (1.1) reduces to multi-point boundary value problems for ordinary differential equations which have been studied in many papers, see, for example, [6, 16–18] and the references cited therein. To our knowledge, only a few papers paid attention to multi-point boundary value problems for impulsive functional differential equations.

In this paper, we are concerned with the existence of extreme solutions for the boundary value problem (1.1). The paper is organized as follows. In Section 2, we establish two comparison principles. In Section 3, we consider a linear problem associated to (1.1) and then give a proof for the existence theorem. In Section 4, we first introduce a new concept of lower and upper solutions. By using the method of upper and lower solutions with a monotone iterative technique, we obtain the existence of extreme solutions for the boundary value problem (1.1).

#### 2. Comparison Principles

In the following, we always assume that the following condition is satisfied:

For any given function , we denote We now present main results of this section.

Theorem 2.1. *Assume that satisfies
**
where , , , and constants satisfy
**
Then for .*

*Proof. *Suppose, to the contrary, that for some .

If , then , and
So and is a maximum value.

If , then , and
So and is a maximum value.

Therefore, there is a such that

Suppose that for . From the first inequality of (2.2), we obtain that for . Hence

If , then , it is easy to obtain that is nondecreasing. Since , it follows that () for . From the first inequality of (2.2), we have that when
which is a contradiction.

If , then , or . If , then () for . If , then for .

From the first inequality of (2.2), we have that when
which is a contradiction.

Suppose that there exist such that and . We consider two possible cases.*Case 1 (). *Since there is such that , for and for all . It is easy to obtain that for . If , then , a contradiction. Hence . Let such that , then . From the first inequality of (2.2), we have
Integrating the above inequality from to , we obtain
Hence
and then integrate from to to obtain
From (2.3), we have that This is a contradiction.*Case 2 (). *Let such that . From the first inequality of (2.2), we have
The rest proof is similar to that of Case 1. The proof is complete.

Theorem 2.2. *Assume that holds and satisfies
**
where constants satisfy (2.3), and , then for .*

*Proof. *Assume that then . By Theorem 2.1, .

Assume that then

Put then for all , and

Hence
By Theorem 2.1, for all , which implies that for .

Assume that then

Put then for all , and
Hence

By Theorem 2.1, for all , which implies that for .

Assume that then

Put then for all , and
Hence

By Theorem 2.1, for all , which implies that for . The proof is complete.

#### 3. Linear Problem

In this section, we consider the linear boundary value problem

Theorem 3.1. *Assume that holds, , , and constants satisfy (2.3) with
**
Further suppose that there exist such that ** on ,**Then the boundary value problem (3.1) has one unique solution and for .*

*Proof. *We first show that the solution of (3.1) is unique. Let be the solution of (3.1) and set . Thus,
By Theorem 2.1, we have that for , that is, on . Similarly, one can obtain that on . Hence .

Next, we prove that if is a solution of (3.1), then . Let . From boundary conditions, we have that for all . From and (3.1), we have

By Theorem 2.1, we have that on . Analogously, on .

Finally, we show that the boundary value problem (3.1) has a solution by five steps as follows.*Step 1. *Let . We claim that(1)(2)(3) for

From and , we have

From (3.9)–(3.14), we obtain that Combining (3.9) and (3.10), we obtain that ( 1) and ( 2) hold.

It is easy to see that on . We show that on . Let then From (3.9)–(3.14), we have

By Theorem 2.1, we have that on , that is, on . So ( 3) holds.*Step 2. *Consider the boundary value problem
where . We show that the boundary value problem (3.16) has one unique solution .

It is easy to check that the boundary value problem (3.16) is equivalent to the integral equation:

where

It is easy to see that with norm is a Banach space. Define a mapping by

For any , we have
Since
the inequality (3.2) implies that is a contraction mapping. Thus there exists a unique such that . The boundary value problem (3.16) has a unique solution.*Step 3. *We show that for any , the unique solution of the boundary value problem (3.16) is continuous in and . Let , be the solution of
Then

From (3.23), we have that

Hence*Step 4. *We show that
for any , and , where is unique solution of the boundary value problem (3.16).

Let . From (3.9), (3.11), (3.12), and (3.16), we have that and

By Theorem 2.1, we obtain that on , that is, on . Similarly, on .*Step 5. *Let . Define a mapping by
where is unique solution of the boundary value problem (3.16). From Step 4, we have
Since is a compact convex set and is continuous, it follows by Schauder’s fixed point theorem that has a fixed point such that
Obviously, is unique solution of the boundary value problem (3.1). This completes the proof.

#### 4. Main Results

Let , . We first give the following definition.

*Definition 4.1. *A function is called a lower solution of the boundary value problem (1.2) if

*Definition 4.2. *A function is called an upper solution of the boundary value problem (1.2) if
Our main result is the following theorem.

Theorem 4.3. *Assume that holds. If the following conditions are satisfied: ** are lower and upper solutions for boundary value problem (1.2) respectively, and on ,**the constants in definition of upper and lower solutions satisfy (2.3), (3.2), and
for .**Then, there exist monotone sequences with such that uniformly on , and are the minimal and the maximal solutions of (1.2), respectively, such that **
on , where is any solution of (1.1) such that on .*

*Proof. *Let . For any , we consider the boundary value problem

Since is a lower solution of (1.2), from , we have that

Similarly, we have that

By Theorem 3.1, the boundary value problem (4.5) has a unique solution . We define an operator by , then is an operator from to .

We will show that

(a)(b) is nondecreasing in .

From and , we have that (a) holds. To prove (b), we show that if .

Let ,and , then by and boundary conditions, we have that

By Theorem 2.1, on , which implies that .

Define the sequences with such that for From (a) and (b), we have

on , and each satisfies
Therefore, there exist such that such that , uniformly on . Clearly, are solutions of (1.1).

Finally, we prove that if is any solution of (1.1), then on . To this end, we assume, without loss of generality, that for some . Since , from property (b), we can obtain

Hence, we can conclude that
Passing to the limit as , we obtain
This completes the proof.

#### Acknowledgments

This work is supported by the NNSF of China (10571050;10871062) and Hunan Provincial Natural Science Foundation of China (NO:09JJ3010), and Science Research Fund of Hunan provincial Education Department (No: 06C052 ).

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#### Copyright

Copyright © 2009 Xuxin Yang 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.