Periodic boundary value problems for second-order impulsive integro-differential equations with integral jump conditions

This paper is concerned with the existence of extremal solutions of periodic boundary value problems for second-order impulsive integro-differential equations with integral jump conditions. We introduce a new definition of lower and upper solutions with integral jump conditions and prove some new maximum principles. The method of lower and upper solutions and the monotone iterative technique are used.

MSC: 34B37, 34K10, 34K45.

Differential equations which have impulse effects describe many evolution processes that abruptly change their state at a certain moment. In recent years, impulsive differential equations have become more important tools in some mathematical models of real processes and phenomena studied in physics, biotechnology, chemical technology, population dynamics and economics; see [1-5]. Many papers have been published about existence analysis of periodic boundary value problems of first and second order for impulsive ordinary or functional or integro-differential equations. We refer the readers to the papers [6-29]. More recent works on existence results of impulsive problems with integral boundary conditions can be found in [30-35] and the reference therein. This literature has lead to significant development of a general theory for impulsive differential equations.

The monotone iterative technique coupled with the method of upper and lower solutions has been used to study the existence of extremal solutions of periodic boundary value problems for second-order impulsive equations; see, for example, [36-41]. This method has been also used to study abstract nonlinear problems; see [42]. However, in most of these papers concerned with applications of the monotone iterative technique to second-order periodic boundary value problems with impulses, the authors assume that the jump conditions at impulse point of solution values and the derivative of solution values depend on the left-hand limits of solutions or the slope of solutions themselves, such as , , , .

In this paper, we consider the periodic boundary value problem for second-order impulsive integro-differential equation (PBVP) with integral jump conditions:

where , is continuous everywhere except at , , exist, , , , , , , , ,

, , , , , .

In [43,44], the authors discussed some kinds of first-order impulsive problems with the integral jump condition

where , , . We note that the jump condition (1.2) depends on functionals of path history before impulse points and after the past impulse points . The aim of our research is to deal with the integral jump conditions

where , , . The integral jump condition (1.3) means that a sudden change of solution values and the derivative of solution values at impulse point depend on the area under the curves of and between to and to , respectively. It should be noticed that the impulsive effects of PBVP (1.1) have memory of the past states.

This paper is organized as follows. Firstly, we introduce a new concept of lower and upper solutions. After that, we establish some new comparison principles and discuss the existence and uniqueness of the solutions for second-order impulsive integro-differential equations with integral jump conditions. By using the method of upper and lower solutions and the monotone iterative technique, we obtain the existence of an extreme solution of PBVP (1.1). Finally, we give an example to illustrate the obtained results.

Let , , for . Let , and . and are Banach spaces with the norms and . Let . A function is called a solution of PBVP (1.1) if it satisfies (1.1).

Definition 2.1 We say that the functions are lower and upper solutions of PBVP (1.1), respectively, if there exist , , , , , , , such that

where

and

where

Now we are in the position to establish some new comparison principles which play an important role in the monotone iterative technique.

Lemma 2.1Assume thatsatisfies

where, , , , are constants and, , , and they satisfy

Then, .

Proof Suppose, to the contrary, that for some . We divide the proof into two cases:

Case (i). There exists a such that and for all .

From (2.1), we have for . Since , then is nondecreasing in and so . However, by (2.1) , then , which implies for all . Thus, , a contradiction.

Case (ii). There exists such that , .

Let , then there exists , for some , such that or . Without loss of generality, we only consider . For the case the proof is similar. It follows that

If for all , then , . Hence, is strictly increasing on J, which contradicts . Then there exists a such that .

Let , . By mean value theorem, we have

Summing up the above inequalities, we obtain

Let , . If by using the method to get (2.3), then we have

If , then the above method together with (2.1), (2.3) implies that

Thus,

Let for some . We first assume that , then . By the mean value theorem, we have

Summing up, we get

Hence,

which contradicts (2.2).

For the case , the proof is similar, and thus we omit it. This completes the proof. □

Lemma 2.2Assume thatsatisfies

where, , , , are constants and, , , and they satisfy (2.2). Thenfor all.

Proof Let , , and define

Note that , for . If we prove that , then and the proof is complete. Since , then we get

Hence, . Indeed, for ,

and

Meanwhile, for , ,

Then by Lemma 2.1, we get for all , which implies that , . □

Consider the linear PBVP

where constants , , , , , , are constants and , , , .

Lemma 2.3is a solution of (2.4) if and only ifis a solution of the following impulsive integral equation:

where

This proof is similar to the proof of Lemma 2.1 in [36], and we omit it.

Lemma 2.4Let, , , , are constants and, , . If

then (2.4) has a unique solutionxinE.

Proof For any , we define an operator F by

where , are given by Lemma 2.3. Then and

By computing directly, we have

and

On the other hand, for , we have

Similarly,

Thus,

By the Banach fixed-point theorem, F has a unique fixed point , and by Lemma 2.3, is also the unique solution of (2.4). This completes the proof. □

In this section, we establish existence criteria for solutions of PBVP (1.1) by the method of lower and upper solutions and the monotone iterative technique. For , we write if for all . In such a case, we denote .

Theorem 3.1Suppose that the following conditions hold:

(H₁) andare lower and upper solutions for PBVP (1.1), respectively, such that.

(H₂) The functionfsatisfies

for all, , , .

(H₃) , , , , are constants, and, , , and they satisfy (2.2), (2.6) and (2.7).

(H₄) The functions, satisfy

where, , , .

Then there exist monotone sequenceswhich converge inEto the extreme solutions of PBVP (1.1) in, respectively.

Proof For any , we consider linear PBVP (2.4) with

By Lemma 2.4, PBVP (2.4) has a unique solution . We define an operator A from to E by . We complete the proof in four steps.

Step 1. We claim that and . We only prove since the second inequality can be proved in a similar manner.

Let and . Then satisfies

We finish Step 1 in two cases.

Case 1. , which implies that

As is a lower solution of PBVP (1.1), then for ,

and

Then by Lemma 2.1, , which implies that , i.e., .

Case 2. , which implies that

Hence,

and

and

, and

Then by Lemma 2.2, , which implies , i.e., .

Step 2. We prove that if , then .

Let , , and , then for , and by (H₂), we obtain

From (H₃), we obtain

Applying Lemma 2.1, we get , which implies .

Step 3. We show that PBVP (1.1) has solutions.

Let , ,  . Following the first two steps, we have

Obviously, each , () satisfies

and

Thus, there exist and such that

Clearly, , satisfy PBVP (1.1).

Step 4. We show that , are extreme solutions of PBVP (1.1).

Let be any solution of PBVP (1.1), which satisfies , . Suppose that there exists a positive integer n such that for , . Setting , then for ,

and

and

and

Still by Lemma 2.1, we have for all , , i.e., . Similarly, we can prove that , . Therefore, , for all , which implies . The proof is complete. □

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

Example 4.1 Consider the following PBVP:

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

Let

we have

where , , , . It is easy to see that

and

whenever , .

Taking , , , , , it follows that

and

Therefore, (4.1) satisfies all the conditions of Theorem 3.1. So, PBVP (4.1) has minimal and maximal solutions in the segment .

Substituting , into monotone iterative scheme, we obtain

and

After using the variational iteration method [45] for (4.2), (4.3), the approximate solutions for and can be illustrated as Figure 1 and Figure 2, respectively.

The authors declare that they have no competing interests.

All authors contributed equally in this article. They read and approved the final manuscript.

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

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