New Existence Results and Comparison Principles for Impulsive Integral Boundary Value Problem with Lower and Upper Solutions in Reversed Order

Abstract

This paper investigates the existence of the extremal solutions to the integral boundary value problem for first-order impulsive functional integrodifferential equations with deviating arguments under the assumption of existing upper and lower solutions in the reversed order. The sufficient conditions for the existence of solutions were obtained by establishing several new comparison principles and using the monotone iterative technique. At last, a concrete example is presented and solved to illustrate the obtained results.

1. Introduction

Impulsive differential equations arise naturally from a wide variety of applications, such as control theory, physics, chemistry, population dynamics, biotechnology, industrial robotic, and optimal control ([1–4]). Therefore, it is very important to develop a general theory for differential equations with impulses including some basic aspects of this theory.

In this paper, we consider the following integral boundary value problem for first-order impulsive functional integrodifferential equations with deviating arguments:

(1.1)

where , , , , , , , , , where and denote the right and the left limits of at , respectively, and

(1.2)

here , , , , , . Let  :  is continuous at , left continuous at and exists, and is continuously differentiable at , and exist, . Evidently, and are Banach spaces with respective norms

(1.3)

In recent years, attention has been given to integral type of boundary conditions. The interest in the study of integral boundary conditions lies in the fact that it has various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, and population dynamics. For a detailed description of the integral boundary conditions, we refer the reader to some recent papers ([5–12]) and the references therein.

The method of upper and lower solutions coupled with its associated monotone iteration scheme is an interesting and powerful mechanism that offers the theoretical as well constructive existence results for nonlinear problem in a closed set, generated by the lower and upper solutions (see [9–26]). In the above-mentioned papers, main results are formulated and proved under the assumption of existing upper and lower solutions in the usual order.

However in many cases, the lower and upper solutions occur in the reversed order. This is a fundamentally different situation. In 2009, Wang et al. [27] successfully investigated boundary value problem for functional differential equations without impulses under the assumption of existing upper and lower solutions in the reversed order. In our recent work [28], the monotone iterative technique, combining with the upper and lower solutions in the reversed order, has been successfully applied to obtain the existence of the extremal solutions for a class of nonlinear first-order impulsive functional differential equations. About other existence results for the nonordered case, see ([29–33]).

Motivated by the above-mentioned works, in this paper, we study the integral boundary value problem (1.1). As far as I am concerned, no paper has considered first-order impulsive functional integrodifferential equations with integral boundary conditions and deviating arguments (i.e., problem (1.1)) under the assumption of existing upper and lower solutions in the reverse order. This paper fills this gap in the literature.

The rest of the paper is organized as follows. In Section 2, we establish several new comparison principles, which play an important role in the proof of main results. Further, to study the nonlinear problem (1.1), we consider the associated linear problem and obtain the uniqueness of the solutions to the associated linear problem. In Section 3, the main theorems are formulated and proved. In Section 4, we give an example about integral boundary value problem for impulsive functional integrodifferential equations of mixed type (1.1).

2. Several Comparison Principles and Linear Problem

Lemma 2.1 (comparison result).

Assume that satisfies

(2.1)

where , , , satisfy

  ,

  ;

here

(2.2)

Then , .

Proof.

Supposing that contrary (i.e., for some ), we consider the following two possible cases:

(1) for all ;

(2)there exist such that and .

Let ; we have

(2.3)

Case 1.

Equation (2.3) implies that for and , hence, is nondecreasing on . By (2.3), we can get

(2.4)

Integrating the above inequality from 0 to , we have

(2.5)

Thus,

(2.6)

Noting condition (i), we have . Besides, , that is,. Since is nondecreasing on , then we have , for all . That is, , for all .

Case 2.

Firstly, we consider (2.3). Let , then , and for some , there exists a , such that or . We only consider , for the case , and the proof is similar.

By (2.3), we have

(2.7)

Let in (2.7); we have

(2.8)

So,

(2.9)

On the other hand,

(2.10)

Let in (2.10), then

(2.11)

That is,

(2.12)

By (2.3), we have

(2.13)

Thus, by (2.9), (2.13), and , we obtain

(2.14)

So, , which contradicts condition (ii). Hence, on .

The proof of Lemma 2.1 is complete.

Corollary 2.2.

Assume that , , , , and condition (ii) in Lemma 2.1 hold. Let satisfy (2.1). Then , .

Proof.

The proof of Corollary 2.2 is easy, so we omit it.

Lemma 2.3 (comparison result).

Let satisfy (2.1). Assume that , , and condition (i) in Lemma 2.1 hold. In addition assume that

  .

Then , .

Proof.

The proof is similar to the proof of Lemma 2.1 [28], so we omit it.

Corollary 2.4.

Assume that , , , , and condition (iii) in Lemma 2.3 hold. Let satisfy (2.1). Then , .

Proof.

The proof of Corollary 2.4 is easy, so we omit it.

Remark 2.5.

Corollary 2.4 holds for if we delete .

Remark 2.6.

In the special case where (2.1) does not contain the operators and , Lemmas 2.1 and 2.3 develop Lemma 2.1 [28], and Corollaries 2.2 and 2.4 develop Corollary 2.1 [28]. Moreover, the condition in Lemma 2.1 and Corollary 2.2 is more extensive than the corresponding condition in [28], and if we let in Lemma 2.3 and Corollary 2.4, we can obtain Lemma 2.1 and Corollary 2.1 in [28], respectively. Therefore, our comparison results in this paper develop and generalize the corresponding results in [28].

To study the nonlinear problem (1.1), we first consider the associated linear problem

(2.15)

where , .

Definition 2.7.

One says is a solution of (2.15) if it satisfies (2.15).

Definition 2.8.

One says that is called a lower solution of (2.15) if

(2.16)

and it is an upper solution of (2.15) if the above inequalities are reversed.

Lemma 2.9.

Let all assumptions of Lemma 2.1 hold. In addition assume that are lower and upper solutions of (2.15), respectively, and , for all . Then the problem (2.15) has a unique solution .

Proof.

The proof is similar to the proof of Lemma 2.2 [28], so we omit it.

Remark 2.10.

In Lemma 2.9, if we replace "Lemma 2.1" by any of "Corollary 2.2", "Lemma 2.3", or "Corollary 2.4", then the conclusion of Lemma 2.9 holds.

3. Nonlinear Problem

Definition 3.1.

One says is a solution of (1.1) if it satisfies (1.1).

Definition 3.2.

One says that is called a lower solution of (1.1) if

(3.1)

and it is an upper solution of (1.1) if the above inequalities are reversed.

Theorem 3.3.

Let all assumptions of Lemma 2.1 hold. In addition assume that

  are lower and upper solutions of (1.1), respectively, and , for all ;

  the function satisfies

(3.2)

for , , , , for all ;

  the function satisfies

(3.3)

for , ;

  there exists such that

(3.4)

if .

Then there exist monotone iterative sequences , , which converge uniformly on to the extremal solutions of (1.1) in  : .

Proof.

For any , we consider the problem

(3.5)

where

(3.6)

Firstly, we verify that , are lower and upper solutions in the reversed order of (3.5). By , we obtain, for ,

(3.7)

and, analogously,

(3.8)

Besides, for ,

(3.9)

In addition,

(3.10)

Therefore, , are lower and upper solutions in the reversed order of (3.5). By Lemma 2.9, we know that (3.5) has a unique solution .

Now, we prove that . Let ; we can get

(3.11)

By Lemma 2.1, we have that , for all . That is, . Similarly, we can show that . Therefore, we have .

Next, we denote an operator by . We prove that is nondecreasing. Let such that . Setting , by , we have

(3.12)

By Lemma 2.1, we know on , that is, is nondecreasing.

Now, let , , , then we have

(3.13)

Obviously, satisfy

(3.14)

with defined by

(3.15)

Therefore, there exist , such that

(3.16)

uniformly on , and the limit functions , satisfy (1.1). Moreover, .

Finally, we prove that , are the extremal solutions of (1.1) in . Let be any solution of (1.1), then . By and the properties of , we have

(3.17)

Thus, taking limit in (3.17) as , we have . That is, , are the extremal solutions of (1.1) in .

The proof of Theorem 3.3 is complete.

Theorem 3.4.

Let conditions and all assumptions of any of Corollary 2.2, Lemma 2.3, or Corollary 2.4 satisfy, then the conclusion of Theorem 3.3 hold.

Proof.

The proof is similar to the proof of Theorem 3.3, so we omit it.

4. Example

Consider the integral boundary value problem

(4.1)

where , , , , , , , for all .

Obviously, , are lower and upper solutions of (4.1), respectively, and .

Note that , , and .

We have

(4.2)

where , , , , for all .

For , , , , , , it is easy to verify that all conditions of Theorem 3.3 hold. Therefore, by Theorem 3.3, there exist monotone iterative sequences , , which converge uniformly on to the extremal solutions of (4.1) in .

Remark 4.1.

For appropriate and suitable choices of , , , and , we see that problem (4.1) has a very general form. For example, we can take , ,, and .

5. Conclusions

In this paper, we have discussed the integral boundary value problem for first-order impulsive functional integrodifferential equations with deviating arguments under the assumption of existing upper and lower solutions in the reversed order. The main results (Theorems 3.3 and 3.4) are new and the following results appear as its special cases.

(i)If we take in (1.1), we obtain the first-order impulsive ordinary integrodifferential equations with integral boundary conditions.

(ii)By taking and in (1.1), our result corresponds to periodic boundary value problem for first-order impulsive functional integrodifferential equations with deviating arguments.

(iii)For , , in (1.1), we get the integral boundary value problem for first-order mixed type integrodifferential equations with deviating arguments.