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Mixed Monotone Iterative Technique for Impulsive Periodic Boundary Value Problems in Banach Spaces
Boundary Value Problems volume 2011, Article number: 421261 (2011)
Abstract
This paper deals with the existence of -quasi-solutions 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 first-order 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 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).
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:
-
(H1)
There exist constants
and
such that
(3.1)for any
and
,
.
-
(H2)
The impulsive function
satisfies
(3.2)for any
and
,
-
(H3)
There exist a constant
such that
(3.3)for all
and increasing or decreasing monotonic sequences
and
-
(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

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 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

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:
-
(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

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.
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Acknowledgments
This paper was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNU-KJCXGC-3-47.
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Chen, P. Mixed Monotone Iterative Technique for Impulsive Periodic Boundary Value Problems in Banach Spaces. Bound Value Probl 2011, 421261 (2011). https://doi.org/10.1155/2011/421261
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DOI: https://doi.org/10.1155/2011/421261