- Research Article
- Open Access
- Published:
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 
, 
, 
and 
such that
and 
, 
.
satisfies
and 
, 
such that
and increasing or decreasing monotonic sequences 
and 
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 
, 
and 
such that
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.
References
- 1.
Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.
- 2.
Guo DJ, Liu X: Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces. Journal of Mathematical Analysis and Applications 1993, 177(2):538-552. 10.1006/jmaa.1993.1276
- 3.
Li Y, Liu Z: Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(1):83-92. 10.1016/j.na.2005.11.013
- 4.
Lakshmikantham V, Leela S: Existence and monotone method for periodic solutions of first-order differential equations. Journal of Mathematical Analysis and Applications 1983, 91(1):237-243. 10.1016/0022-247X(83)90102-6
- 5.
He Z, Yu J: Periodic boundary value problem for first-order impulsive ordinary differential equations. Journal of Mathematical Analysis and Applications 2002, 272(1):67-78. 10.1016/S0022-247X(02)00133-6
- 6.
Lakshmikantham V, Leela S, Vatsala AS: Method of quasi-upper and lower solutions in abstract cones. Nonlinear Analysis 1982, 6(8):833-838. 10.1016/0362-546X(82)90067-0
- 7.
Guo DJ, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Analysis: Theory, Methods & Applications 1987, 11(5):623-632. 10.1016/0362-546X(87)90077-0
- 8.
Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.
- 9.
Heinz H-P: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Analysis: Theory, Methods & Applications 1983, 7(12):1351-1371. 10.1016/0362-546X(83)90006-8
- 10.
Li YX: Existence of solutions to initial value problems for abstract semilinear evolution equations. Acta Mathematica Sinica 2005, 48(6):1089-1094.
- 11.
Guo DJ, Sun JX: Ordinary Differential Equations in Abstract Spaces. Shandong Science and Technology, Jinan, China; 1989.
- 12.
Du YH: Fixed points of increasing operators in ordered Banach spaces and applications. Applicable Analysis 1990, 38(1-2):1-20. 10.1080/00036819008839957
Acknowledgments
This paper was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNU-KJCXGC-3-47.
Author information
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Accepted:
Published:
Keywords
- Banach Space
- Impulsive Function
- Impulsive Differential Equation
- Couple Fixed Point
- Order Banach Space
