This paper deals with the existence of quasisolutions for impulsive periodic boundary value problems in an ordered Banach space . Under a new concept of upper and lower solutions, a new monotone iterative technique on periodic boundary value problems of impulsive differential equations has been established. Our result improves and extends some relevant results in abstract differential equations.
1. Introduction
The theory of impulsive differential equations is a new and important branch of differential equation theory, which has an extensive physical, chemical, biological, and engineering background and realistic mathematical model, and hence has been emerging as an important area of investigation in the last few decades; see [1]. Correspondingly, applications of the theory of impulsive differential equations to different areas were considered by many authors, and some basic results on impulsive differential equations have been obtained; see [2–5]. But many of them are about impulsive initial value problem; see [2, 3] and the references therein. The research on impulsive periodic boundary value problems is seldom; see [4, 5].
In this paper, we use a monotone iterative technique in the presence of coupled lower and upper quasisolutions to discuss the existence of solutions to the impulsive periodic boundary value problem (IPBVP) in an ordered Banach space
where ,,; ; is an impulsive function, . denotes the jump of at that is, where and represent the right and left limits of at , respectively.
The monotone iterative technique in the presence of lower and upper solutions is an important method for seeking solutions of differential equations in abstract spaces. Early on, Lakshmikantham and Leela [4] built a monotone iterative method for the periodic boundary value problem of firstorder differential equation in
and they proved that, if PBVP(1.2) has a lower solution and an upper solution with and nonlinear term satisfies the monoton condition
with a positive constant , then PBVP(1.2) has minimal and maximal solutions between and which can be obtained by a monotone iterative procedure starting from and , respectively. Later, He and Yu [5] developed the problem to impulsive differential equation
where , , ,
But all of these results are in real spaces We not only consider problems in Banach spaces, but also expand the nonlinear term to the case of If is nondecreasing in and is nonincreasing in then the monotonity condition (1.3) is not satisfied, and the results in [4, 5] are not right, in this case, we studied the IPBVP(1.1). As far as we know, no work has been done for the existence of solutions for IPBVP(1.1) in Banach spaces.
In order to apply the monotone iterative technique to the initial value problem without impulse
Lakshmikantham et al. [6] and Guo and Lakshmikantham [7] obtained the existence of coupled quasisolutions of problem (1.5) by mixed monotone sequence of coupled quasiupper and lower solutions under the concept of quasiupper and lower solutions. In this paper, we improve and extend the abovementioned results, and obtain the existence of the coupled minimal and maximal quasisolutions and the solutions between the coupled minimal and maximal quasisolutions of the problem (1.1) through the mixed monotone iterative about the coupled quasisolutions. If the coupled upper and lower quasisolutions are equivalent to coupled upper and lower quasisolutions of the IPBVP(1.1). If , and the coupled upper and lower quasisolutions are equivalent to upper and lower solutions of IPBVP(1.4).
2. Preliminaries
Let be an ordered Banach space with the norm and partial order whose positive cone is normal with normal constant Let , is a constant; ; , , Let is continuous at , and left continuous at , and exists, Evidently, is a Banach space with the norm . An abstract function is called a solution of IPBVP(1.1) if satisfies all the equalities of (1.1)
Let and exist, . For it is easy to see that the left derivative of at exists and and set , then If is a solution of IPBVP(1.1), by the continuity of
Let denote the Banach space of all continuous value functions on interval with the norm . Let denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [8]. For any and set If is bounded in then is bounded in and
Now, we first give the following lemmas in order to prove our main results.
Lemma 2.1 (see [9]).
Let be a bounded and countable set. Then is Lebesgue integral on and
Lemma 2.2 (see [10]).
Let be bounded. Then exist a countable set , such that
Lemma 2.3 (see [11]).
Let be equicontinuous. Then is continuous on and
Lemma 2.4 (see [8]).
Let be a Banach space and is a bounded convex closed set in be condensing, then has a fixed point in
To prove our main results, for any we consider the periodic boundary value problem (PBVP) of linear impulsive differential equation in
where , ,
Lemma 2.5.
For any , and , the linear PBVP(2.3) has a unique solution given by
where
Proof.
For any , and , the linear initial value problem
has a unique solution given by
where is a constant [3].
If is a solution of the linear initial value problem (2.5) satisfies namely
then it is the solution of the linear PBVP(2.3). From (2.7), we have
So, (2.4) is satisfied.
Inversely, we can verify directly that the function defined by (2.4) is a solution of the linear PBVP(2.3). Therefore, the conclusion of Lemma 2.5 holds.
Definition 2.6.
Let be a constant. If functions satisfy
we call , coupled lower and upper quasisolutions of the IPBVP(1.1). Only choose "" in (2.9) and (2.10), we call coupled quasisolution pair of the IPBVP(1.1). Furthermore, if we call a solution of the IPBVP(1.1).
Now, we define an operator as following:
where
Evidently, is also an ordered Banach space with the partial order "" reduced by the positive cone . is also normal with the same normal constant . For with we use to denote the order interval in and to denote the order interval in .
3. Main Results
Theorem 3.1.
Let be an ordered Banach space, whose positive cone is normal, and , . Assume that the IPBVP(1.1) has coupled lower and upper quasisolutions and with . Suppose that the following conditions are satisfied:
There exist constants and such that
for any and , .
The impulsive function satisfies
for any and ,
There exist a constant such that
for all and increasing or decreasing monotonic sequences and
The sequences and are convergent, where , ,
Then the IPBVP(1.1) has minimal and maximal coupled quasisolutions between and which can be obtained by a monotone iterative procedure starting from and , respectively.
Proof.
By the definition of and Lemma 2.5, is continuous, and the coupled quasisolutions of the IPBVP(1.1) is equivalent to the coupled fixed point of operator Combining this with the assumptions and , we know is a mixed monotone operator (about the mixed monotone operator, please see [6, 7]).
Next, we show , . Let by (2.9), and , By Lemma 2.5
namely, . Similarly, it can be show that . So,
Now, we define two sequences and in by the iterative scheme
Then from the mixed monotonicity of , it follows that
We prove that and are uniformly convergent in
For convenience, let , , , and . Since, and by (2.11) and the boundedness of and we easy see that and is equicontinuous in every interval so, is equicontinuous in every interval where , , From and it follows that and for Let , by Lemma 2.3, . Going from to interval by interval we show that in
For from (2.11), using Lemma 2.1 and assumption and we have
Hence by the Belman inequality, in In particular, , this means that and are precompact in Thus and are precompact in and ,
Now, for by (2.11) and the above argument for we have
Again by Belman inequality, in from which we obtain that , and ,
Continuing such a process interval by intervai up to we can prove that in every ,
For any if we modify the value of , at via , , then and it is equicontinuous. Since , is precompact in for every By the ArzelaAscoli theorem, is precompact in Hence, has a convergent subsequence in Combining this with the monotonicity (3.6), we easily prove that itself is convergent in In particular, is uniformly convergent over the whole of Hence, is uniformly convergent in Set
Letting in (3.5) and (3.6), we see that and , By the mixed monotonicity of it is easy to see that and are the minimal and maximal coupled fixed points of in and therefore, they are the minimal and maximal coupled quasisolutions of the IPBVP(1.1) in respectively.
In Theorem 3.1, if is weakly sequentially complete, condition and hold automatically. In fact, by Theorem in [12], any monotonic and orderbounded sequence is precompact. By the monotonicity (3.6) and the same method in proof of Theorem 3.1, we can easily see that and are convergent on In particular, and are convergent. So, condition holds. Let and be increasing or decreasing sequences obeying condition then by condition , is a monotonic and orderbounded sequence, so Hence, condition holds. From Theorem 3.1, we obtain the following corollary.
Corollary 3.2.
Let be an ordered and weakly sequentially complete Banach space, whose positive cone is normal, and , If the IPBVP(1.1) has coupled lower and upper quasisolutions and with and conditions and are satisfied, then the IPBVP(1.1) has minimal and maximal coupled quasisolutions between and which can be obtained by a monotone iterative procedure starting from and respectively.
If we replace the assumption by the following assumption:
There exist positive constants and such that
for any and ,
We have the following result.
Theorem 3.3.
Let be an ordered Banach space, whose positive cone is normal, and , If the IPBVP(1.1) has coupled lower and upper quasisolutions and with and conditions , , and hold, then the IPBVP(1.1) has minimal and maximal coupled quasisolutions between and which can be obtained by a monotone iterative procedure starting from and respectively.
Proof.
For let be a increasing sequence and be a decreasing sequence. For with by and
By this and the normality of cone we have
From this inequality and the definition of the measure noncompactness, it follows that
where If is a increasing sequence and is a decreasing sequence, the above inequality is also valid. Hence holds.
Therefore, by Theorem 3.1, the IPBVP(1.1) has minimal and maximal coupled quasisolutions between and which can be obtained by a monotone iterative procedure starting from and , respectively.
Now, we discuss the existence of the solution to the IPBVP(1.1) between the minimal and maximal coupled quasisolutions and If we replace the assumptions and by the following assumptions:
The impulsive function satisfies
for any and , and there exist , such that
for any countable sets and in
There exist a constant such that
for any where and are countable sets in
We have the following existence result.
Theorem 3.4.
Let be an ordered Banach space, whose positive cone is normal, and , If the IPBVP(1.1) has coupled lower and upper quasisolutions and with such that assumptions , , and hold, then the IPBVP(1.1) has minimal and maximal coupled quasisolutions and between and and at least has one solution between and
Proof.
We can easily see that , Hence, by the Theorem 3.1, the IPBVP(1.1) has minimal and maximal coupled quasisolutions and between and Next, we prove the existence of the solution of the equation between and Let clearly, is continuous and the solution of the IPBVP(1.1) is equivalent to the fixed point of operator Since is bounded and equicontinuous for any by Lemma 2.2, there exist a countable set such that
By assumptions and and Lemma 2.1,
Since is equicontinuous, by Lemma 2.3, . Combing (3.17) and .
We have
Hence, the operator is condensing, by the Lemma 2.4, has fixed point in
Lastly, since , by the mixed monotonity of
Similarly, in general, letting we get Therefore, the IPBVP(1.1) at least has one solution between and
Remark 3.5.
If and then Theorems 3.1, 3.3 and 3.4 are generalizations of the main results of [5] in Banach spaces.
Remark 3.6.
If and then Theorems 3.1, 3.3 and 3.4 are generalizations of the Theory of [4] in Banach spaces.
4. An Example
Consider the PBVP of infinite system for nonlinear impulsive differential equations:
4.1. Conclusion
IPBVP(4.1) has minimal and maximal coupled quasisolutions.
Proof.
Let , with norm and Then is a weakly sequentially complete Banach space and is normal cone in IPBVP(4.1) can be regarded as an PBVP of the form (1.1) in In this case, , , and in which
, and ,
Evidently, , Let
, Then it is easy to verify that , are coupled lower and upper quasisolutions of the IPBVP(4.1), and conditions , hold. Hence, our conclusion follows from Corollary 3.2.
Acknowledgments
This paper was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNUKJCXGC347.
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