Abstract and Applied Analysis

Volume 2013 (2013), Article ID 134691, 9 pages

http://dx.doi.org/10.1155/2013/134691

Research Article

Existence Result for Impulsive Differential Equations with Integral Boundary Conditions

Peipei Ning, Qian Huan, and Wei Ding

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 5 December 2012; Accepted 5 January 2013

Academic Editor: Yonghui Xia

Copyright © 2013 Peipei Ning 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 investigate the following differential equations: , with impulsive and integral boundary conditions , , , , where . The expression of Green's function and the existence of positive solution for the system are obtained. Upper and lower bounds for positive solutions are also given. When , , , and take different values, the system can be simplified to some forms which has been studied in the works by Guo and LakshmiKantham (1988), Guo et al. (1995), Boucherif (2009), He et al. (2011), and Atici and Guseinov (2001). Our discussion is based on the fixed point index theory in cones.

1. Introduction

The theory of impulsive differential equations in abstract spaces has become a new important branch and has developed rapidly (see [1–4]). As an important aspect, impulsive differential equations with boundary value problems have gained more attention. In recent years, experiments in a variety of different areas (especially in applied mathematics and physics) show that integral boundary conditions can represent the model more accurately. And researchers have obtained many good results in this field.

In this paper, we study the existence of positive solutions for the following system:

where , , , , . , are left continuous at , . . And , , are continuous and positive functions.

When , , , and   take different values, the system can be simplified to some forms which have been studied. For example, [5–10] discussed the existence of positive solution in case .

Let , , [11, 12] investigated the system with only one impulse. Reference [13] studied the system when , . Readers can read the papers in [13] for details.

Throughout the rest of the paper, we assume is a fixed positive number, and is a parameter. , are real-valued measurable functions defined on , and they satisfy the following condition:

(H1) , , almost everywhere, and

This paper aims to obtain the positive solution for (1). In Section 2, we introduce some lemmas and notations. In particular, the expression and some properties of Green's functions are investigated. After the preparatory work, we draw the main results in Section 3.

2. Preliminaries

Theorem 1 (Krasnoselskii's fixed point theorem). Let be a Banach space and . Assume , are open sets in with , and be a completely continuous operator such that either

(i), , and , ; or (ii), , and , .

Then has a fixed point in .

Definition 2. For two differential functions and , we defined their Wronskian by

Consider the linear nonhomogeneous problem of the form

Its corresponding homogeneous equation is

Lemma 3. Suppose that and form a fundamental set of solutions for the homogeneous problem (5). Then the general solution of the nonhomogeneous problem (4) is given by

where and are arbitrary constants.

Proof. We just need to show that the function

is a particular solution of (4). From (7), we have for ,

Besides, from (7) and (8), we have

Thus, satisfies (4).

Consider the following boundary value problem with integral boundary conditions:

Denote by and the solutions of the homogenous equation (5) satisfying the initial conditions

(H2) Let , denote a function

satisfies .

For convenience, we denote , .

Lemma 4. Let be a nonnegative continuous function defined for , and a nonnegative integrable function on . Then for arbitrary nonnegative continuous function defined on , the Volterra integral equation

has a unique solution . Moreover, this solution is continuous and satisfied the inequality

Proof. We solve (14) by the method of successive approximations setting

If the series converges uniformly with respect to , then its sum will be, obviously, a continuous solution of (14). To prove the uniform convergence of this series, we put

Then it is easy to get from (16) that

Hence it follows that (14) has a continuous solution

and because , , , for this solution the inequality (15) holds. Uniqueness of the solution of (14) can be proved in a usual way. The proof is complete.

Remark 5. Evidently, the statement of Lemma 4 is also valid for the Volterra equation of the form

Lemma 6. For the solution of the BVP (11), the formula

holds, where

Proof. By Lemma 3, the general solutions of the nonhomogeneous problem (4) has the form

where and are arbitrary constants. Now we try to choose the constants and so that the function satisfies the boundary conditions of (11).

From (23), we have

Consequently,

Substituting these values of and into the first boundary condition of (11), we find

Similarly from the second boundary condition of (11), we can find

Putting these values of and in (23), we get the formula (21), (22).

Lemma 7. Let condition (H1) hold. Then for the Wronskian of solution and , the inequality , holds.

Proof. Using the initial conditions (12), we can deduce from (5) for and the following equations:

From (28), by condition (H1) and Lemma 4, it follows that

Now from (3), we get , . The proof is complete.

From (21), (22), and Lemma 7, the following lemma follows.

Lemma 8. Under condition (H1) the Green's function of the BVP (11) is positive. That is, for .

Let denote the Banach of all continuous functions equipped with the form , for any . Denote , then is a positive cone in .

Let us set , , and by Lemma 8, obviously, , .

Define a mapping in Banach space by

where

Lemma 9. The fixed point of the mapping is a solution of (1).

Proof. Clearly, is continuous in for . For ,

where

We have

where

We can easy get that

which implies that the fixed poind of is a solution of (1). The proof is complete.

Lemma 10. Let , then is a cone.

Proof. (i) For for all and for all , , we have

Moreover

Thus .

(ii) If and , we have

It implies that . Hence is a cone.

Defined a linear operator by

Then we have the following lemma.

Lemma 11. If (H2) is satisfied, then

(i) is a bounded linear operator, ;(ii) is invertible;(iii).

Proof. (i)

for all , .

Using , it is easy to see that .

Let . Then for all . Since , it follows that for each . So .

(ii) We want to show that is invertible, or equivalently is not an eigenvalue of .

Since , it follows from condition (H2) that .

So

On the other hand, we suppose 1 is an eigenvalue of , then there exists a such that . Moreover, we can obtain that . So . Thus this assumption is false.

Conversely, 1 is not an eigenvalue of . Equivalently, is invertible.

(iii) We use the theory of Fredholm integral equations to find the expression for .

Obviously, for each , .

By (40), we can get

The condition implies that 1 is not an eigenvalue of the kernel . So (43) has a unique continuous solution for every continuous function .

By successive substitutions in (43), we obtain

where the resolvent kernel is given by

Here , and .

The series on the right in (45) is convergent because .

It can be easily verified that .

So we can get

Therefore

So

Thus . This completes the proof of the lemma.

Remark 12. Since for each , it is easy to prove that .

3. Main Results

Consider the following boundary value problem (BVP) with impulses:

Denote a nonlinear operator by

It is easy to see that solutions of (49) are solutions of the following equation:

According to Lemma 11, is a solution of (51) if and only if it is a solution of

It follows from (46) that is a solution of (52) if and only if

So, the operator can be written as

It satisfies the conditions of Theorem 1 with and the cone .

Let us list some marks and conditions for convenience.

The nonlinearity is continuous and satisfies the following.

(H3) There exist and , with such that

for all , .

(H4) There exist and , with such that

for all , .

Then, we can get the following theorem.

Theorem 13. Assume (H1), (H2), (H3), and (H4) are satisfied. And

then, if satisfies

The problem (49) has at least one positive solution.

Proof. First of all, we show that operator is defined by (54) maps into itself. Let .

Then for all that , and

Because from the formula (54), we have

Hence, inequality (59) is established.

This implies that

or equivalently

On the other hand, it follows that

In fact, we have

It follows from (62) that

So, we get

This show that .

It is easy to see that is the complete continuity.

We now proceed with the construction of the open sets and .

First, let with . Inequality (59) implies

By condition (H3) and (58), we obtain

So

Consequently, .

Let . Then, we have for .

Next, let and set .

For with , we have

It follows from (63) that

By condition (H4) and (58), we obtain

Since we have for all . It follows from the above inequality that

Hence for .

It follows from (i) of Theorem 1 that has a fixed point in , and this fixed point is a solution of (49).

This completes the proof.

Next, with and as above, we assume that satisfied the following.

(H5) There exist , with such that

for all , .

(H6) There exist , with such that

for all , .

Theorem 14. Assume (H1), (H2), (H5), and (H6) are satisfied. And

then, if satisfies

The problem (49) has at least one positive solution.

Proof. Let be a completely continuous operator defined by (54). Then maps the cone into itself.

First, let with . Inequality (63) implies

By condition (H5) and (77), we obtain

Hence

Since , we have for all . It follows from the above inequality that

Let . Then, we have for .

Next, let and set .

Then for with for all , we have . Inequality (59) implies

By condition (H6) and (77), we obtain

So

Therefore with .

Then, we have for .

We see the case (ii) of Theorem 1 is met. It follows that has a fixed point in , and this fixed point is a solution of (49).

This completes the proof.

Acknowledgments

This work was supported by the NNSF of China under Grant no. 11271261, Natural Science Foundation of Shanghai (no. 12ZR1421600), Shanghai Municipal Education Commission (no. 10YZ74), the Slovenian Research Agency, and a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Programme (FP7-PEOPLE-2012-IRSES-316338).

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