International Journal of Differential Equations

Volume 2014 (2014), Article ID 791240, 12 pages

http://dx.doi.org/10.1155/2014/791240

Research Article

Periodic Boundary Value Problems for First-Order Impulsive Functional Integrodifferential Equations with Integral-Jump Conditions

Chatthai Thaiprayoon,1,2 Decha Samana,1,2 and Jessada Tariboon2,3

¹Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

²Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road, Bangkok 10400, Thailand

³Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand

Received 6 January 2014; Accepted 11 February 2014; Published 23 March 2014

Academic Editor: Kanishka Perera

Copyright © 2014 Chatthai Thaiprayoon 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.

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive functional integrodifferential equations with integral-jump conditions. An example is also given to illustrate our results.

1. Introduction

The theory of impulsive differential equations is now being recognized as not only being richer than the corresponding theory of differential equations without impulses, but also representing a more natural framework for mathematical modelling of many real world phenomena and applications; see [1–5] and the references therein. Monotone iterative technique coupled with the method of upper and lower solutions has provided an effective mechanism to prove constructive existence results for initial and boundary value problems for nonlinear differential equations; see [6]. However, many papers have studied applications of the monotone iterative technique to impulsive problems; see, for example, [7–15]. In those articles, the authors assumed that , that is, a short-term rapid change of the state (jump condition) at impulse point , depends on the left side of the limit of .

In [16–18] the authors discussed some classes of first-order impulsive problems with the impulsive integral conditions:

where and , . Furthermore, Thaiprayoon et al. [19] have used such technique to investigate the existence criteria of extremal solutions of multipoint impulsive problems to include multipoint jump conditions

for , . Recently, Thiramanus and Tariboon [20] have given some results on impulsive differential inequalities with integral-jump conditions of the form:

where . We note that if , , , then the above inequalities mean that the bound of jump condition at is a functional of past states on the interval before the impulse point .

In spirit of the results from [20], this paper considers the periodic boundary value problem for first-order impulsive functional integrodifferential equation (PBVP) with integral-jump conditions:

where , , , , , ,

, , , , and .

We first introduce a new concept of lower and upper solutions, then establish a new comparison principle, and discuss the existence and uniqueness of the solutions for first-order impulsive functional integrodifferential equations with integral-jump conditions. By using the method of upper and lower solutions and monotone iterative technique, we obtain the existence of extreme solution of PBVP (4). Finally, we give an example to illustrate the obtained results.

2. Preliminaries

Let , , , and be continuous everywhere except for some , at which and exist and , , and let be continuous everywhere except for some at which and exist and , . Clearly, is a Banach space with the norm . Let . A function is called a solution of PBVP (4) if it satisfies (4).

Definition 1. We say that the functions , are lower and upper solutions of PBVP (4) if there exist , , , , , and such that

where

where

Denote . We prove the comparison principle by using the following lemma (see [20]).

Lemma 2. Let , ,and, , , be constants and let , . If

then, for ,

Now we are in the position to establish a new comparison principle, which plays an important role in monotone iterative technique.

Lemma 3. Assume that satisfies

where constants , , , , , and , . If

where , , and are given by Definition 1 with and

then , .

Proof. Case 1. One has . Suppose that there exists such that and distinguish two cases.

Case (a). for all , ; then

so that is nondecreasing in , and then . Since , then is a constant function , which implies that

getting a contradiction.

Case (b). for some . Let ; then there exists , for some such that or . Without loss of generality, we only consider , and for the case the proof is similar.

From (11), it is easy to see that

We consider the inequalities

From Lemma 2, we have

Let in (18); then

so that

If , then (20) with , for all implies

This contradicts the condition (12).

Suppose that . If for , then Lemma 2 provides that

Since

and integrating (11) from into , we obtain

Hence,

We note that if , then .

Let in (25); then

From (26), we have

which gives

contradicting condition (12). For the case , the proof is similar.

Case 2. One has . Set . It follows that , and for ,

and for , ,

In view of Case 1, we see that on , and therefore on . This completes the proof.

Corollary 4. Assume that satisfies

where constants , , , , , and , . If

where

and , are given by Lemma 3, then , for .

Corollary 5. Assume that satisfies

where constants , , , , , and , , and condition (32) holds. Then , for .

Let us consider the following linear problem of PBVP (4):

where , , , , , , , ,  , , and .

Lemma 6. A function is a solution of (35) if and only if is a solution of the following impulsive integral equation:

where and

Proof. If is a solution of (35), by directly integrating, we obtain

If is a solution of the above-mentioned integral equation, then

The proof is complete.

Lemma 7. Let , , , , , , , , , , and and assume that

Then problem (35) has a unique solution in .

Proof. We define the mapping by

for any and is given by Lemma 6. Then

The above result and condition (40) imply that is a contractive mapping, which completes the proof.

Corollary 8. Let , , , , , , , , , , and and assume that

Then problem (35) has a unique solution in .

3. Main Results

In this section, we establish existence criteria for solutions of the PBVP (4) by the method of lower and upper solutions and monotone iterative technique. For , , we write if for all . In such a case, we denote , .

Theorem 9. Assume the existence of lower and upper solutions for PBVP (4) and also suppose that the following conditions hold.

The function satisfies

where , , , and , , where , , , , and .

The functions satisfy

where , , where and , .

If inequalities (12) and (40) hold, then there exists a solution of PBVP (4) such that , for .

Proof. We consider the following modified problem relative to PBVP (4):

where and

If is such that on , then is a solution of PBVP (4) if and only if is a solution of (46). We will show that (46) is solvable and that every solution of (46) satisfies on . Suppose that is a solution of (46). We will show that . Let . Then, we have since and

By Lemma 3, we get on ; that is, . Similar arguments show that .

It remains to prove that (46) possesses at least one solution. By Lemma 6, PBVP (46) is equivalent to the following impulsive integral equation:

where . For any , define a continuous compact operator by

Let , such that , , , and take the compact sets, , , , and . Since is continuous, then we can choose a constant , such that , . For , we see that any solution of

satisfies

From the continuity of , , and on , we can choose some such that , . Then we have

where

Hence, by Schaefer’s theorem [21], we get that has at least a fixed point , which is a solution of (46). Such a solution lies between and and, consequently, is a solution of (4). Thus, the proof is complete.

Theorem 10. Assume that there exist lower and upper solutions for PBVP (4) and assume the following.

The function satisfies

where , , , and , , where , , , , and .

The functions satisfy

where , , where and , .

Suppose that inequalities (12) and (40) hold. Then there exist monotone sequences , with , , which converge uniformly on to the extremal solutions of the periodic boundary value problem (4) in .

Proof. For any , consider PBVP (35) with

By Lemmas 6 and 7, PBVP (35) possesses a unique solution . We define an operator by ; then the operator has the following properties:

(i), ;(ii) for any , with .

First we prove (i). Let , where . Then, we have since and

By Lemma 3, we get on ; that is, . Analogously, we have .

Now, we claim (ii). Set , , where , with . Let ; by -, we have

By Lemma 3, we have on and so . Thus we may define the sequences , by , , , and . From (i), (ii), we obtain

and each , satisfies

where

Hence, there exist , such that and uniformly on . Clearly, , satisfy PBVP (4). We will prove that , are extreme solutions of PBVP (4). Let be any solution of PBVP (4), which satisfies , . Also suppose there exists a positive integer such that for , . Setting , then for ,

According to Lemma 3, we get that on . Similarly, we obtain on . Since on , by induction, we have on for all . Therefore, on by taking limit as . The proof is complete.

4. An Example

In this section, in order to illustrate our results, we consider an example.

Example 11. Consider the following periodic boundary value problem:

where , , , , , , and .

Obviously, , are lower and upper solutions for (64), respectively, and .

Let

We have

where , , , and , . It is easy to see that

where , .

Taking , , , , and , it follows that

Thus,

Therefore, (64) satisfies all conditions of Theorem 10. So PBVP (64) has minimal and maximal solutions in the segment .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand, and King Mongkut’s University of Technology North Bangkok, Thailand.

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