Mixed Monotone Iterative Technique for Impulsive Periodic Boundary Value Problems in Banach Spaces

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

(1.1)

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 first-order differential equation in

(1.2)

and they proved that, if PBVP(1.2) has a lower solution and an upper solution with and nonlinear term satisfies the monoton condition

(1.3)

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

(1.4)

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

(1.5)

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 above-mentioned 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).

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

(2.1)

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

(2.2)

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

(2.3)

where , ,

Lemma 2.5.

For any , and , the linear PBVP(2.3) has a unique solution given by

(2.4)

where

Proof.

For any , and , the linear initial value problem

(2.5)

has a unique solution given by

(2.6)

where is a constant [3].

If is a solution of the linear initial value problem (2.5) satisfies namely

(2.7)

then it is the solution of the linear PBVP(2.3). From (2.7), we have

(2.8)

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

(2.9)

(2.10)

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:

(2.11)

where

(2.12)

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 .

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:

1. (H1)
   There exist constants and such that

   

   (3.1)

   for any and , .

    
2. (H2)
   The impulsive function satisfies

   

   (3.2)

   for any and ,

    
3. (H3)
   There exist a constant such that

   

   (3.3)

   for all and increasing or decreasing monotonic sequences and

    
4. (H4)

   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

(3.4)

namely, . Similarly, it can be show that . So,

Now, we define two sequences and in by the iterative scheme

(3.5)

Then from the mixed monotonicity of , it follows that

(3.6)

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

(3.7)

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

(3.8)

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 Arzela-Ascoli 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

(3.9)

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 order-bounded 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 order-bounded 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:

1. (H5)
   There exist positive constants and such that

   

   (3.10)

   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

(3.11)

By this and the normality of cone  we have

(3.12)

From this inequality and the definition of the measure noncompactness, it follows that

(3.13)

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

(3.14)

for any and , and there exist , such that

(3.15)

for any countable sets and in

There exist a constant such that

(3.16)

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

(3.17)

By assumptions and and Lemma 2.1,

(3.18)

Since is equicontinuous, by Lemma 2.3, . Combing (3.17) and .

We have

(3.19)

Hence, the operator is condensing, by the Lemma 2.4, has fixed point in

Lastly, since , by the mixed monotonity of

(3.20)

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.