Abstract
This paper is concerned with the existence of extremal solutions of periodic boundary value problems for secondorder impulsive integrodifferential 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.
Keywords:
impulsive integrodifferential equation; lower and upper solutions; periodic boundary value problem; monotone iterative technique1 Introduction
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 [15]. Many papers have been published about existence analysis of periodic boundary value problems of first and second order for impulsive ordinary or functional or integrodifferential equations. We refer the readers to the papers [629]. More recent works on existence results of impulsive problems with integral boundary conditions can be found in [3035] 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 secondorder impulsive equations; see, for example, [3641]. 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 secondorder 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 lefthand limits of solutions or the slope of solutions themselves, such as , , , .
In this paper, we consider the periodic boundary value problem for secondorder impulsive integrodifferential equation (PBVP) with integral jump conditions:
where , is continuous everywhere except at , , exist, , , , , , , , ,
In [43,44], the authors discussed some kinds of firstorder 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 secondorder impulsive integrodifferential 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.
2 Preliminaries
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.
where, , , , are constants and, , , and they satisfy
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. □
where, , , , are constants and, , , and they satisfy (2.2). Thenfor all.
Note that , for . If we prove that , then and the proof is complete. Since , then we get
and
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 fixedpoint theorem, F has a unique fixed point , and by Lemma 2.3, is also the unique solution of (2.4). This completes the proof. □
3 Main results
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:
(H_{1}) andare lower and upper solutions for PBVP (1.1), respectively, such that.
(H_{2}) The functionfsatisfies
(H_{3}) , , , , are constants, and, , , and they satisfy (2.2), (2.6) and (2.7).
(H_{4}) The functions, satisfy
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.
We finish Step 1 in two cases.
As is a lower solution of PBVP (1.1), then for ,
and
Then by Lemma 2.1, , which implies that , i.e., .
Hence,
and
and
Then by Lemma 2.2, , which implies , i.e., .
Step 2. We prove that if , then .
Let , , and , then for , and by (H_{2}), we obtain
From (H_{3}), 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. □
4 An example
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
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in this article. They read and approved the final manuscript.
Acknowledgements
This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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