- Research Article
- Open access
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Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces
Journal of Inequalities and Applications volume 2010, Article number: 293410 (2010)
Abstract
By constructing a mixed monotone iterative technique under a new concept of upper and lower solutions, some existence theorems of mild -periodic (
-quasi) solutions for abstract impulsive evolution equations are obtained in ordered Banach spaces. These results partially generalize and extend the relevant results in ordinary differential equations and partial differential equations.
1. Introduction and Main Result
Impulsive differential equations are a basic tool for studying evolution processes of real life phenomena that are subjected to sudden changes at certain instants. In view of multiple applications of the impulsive differential equations, it is necessary to develop the methods for their solvability. Unfortunately, a comparatively small class of impulsive differential equations can be solved analytically. Therefore, it is necessary to establish approximation methods for finding solutions. The monotone iterative technique of Lakshmikantham et al. (see [1–3]) is such a method which can be applied in practice easily. This technique combines the idea of method of upper and lower solutions with appropriate monotone conditions. Recent results by means of monotone iterative method are obtained in [4–7] and the references therein. In this paper, by using a mixed monotone iterative technique in the presence of coupled lower and upper -quasisolutions, we consider the existence of mild
-periodic (
-quasi)solutions for the periodic boundary value problem (PBVP) of impulsive evolution equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ1_HTML.gif)
in an ordered Banach space , where
is a closed linear operator and
generates a
-semigroup
in
;
only satisfies weak Carathé odory condition,
,
is a constant;
;
is an impulsive function,
;
denotes the jump of
at
, that is,
, where
and
represent the right and left limits of
at
, respectively. Let
is continuous at
and left continuous at
, and
exists,
. Evidently,
is a Banach space with the norm
. Let
,
. Denote by
the Banach space generated by
with the norm
. An abstract function
is called a solution of the PBVP(1.1) if
satisfies all the equalities of (1.1).
Let be an ordered Banach space with the norm
and the partial order "
", whose positive cone
is normal with a normal constant
. Let
. If functions
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ2_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ3_HTML.gif)
we call coupled lower and upper
-quasisolutions of the PBVP(1.1). Only choosing "
" in (1.2) and (1.3), we call
coupled
-periodic
-quasisolution pair of the PBVP(1.1). Furthermore, if
, we call
an
-periodic solution of the PBVP(1.1).
Definition 1.1.
Abstract functions are called a coupled mild
-periodic
-quasisolution pair of the PBVP(1.1) if
and
satisfy the following integral equations:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ4_HTML.gif)
where and
for any
,
is an identity operator. If
, then
is called a mild
-periodic solution of the PBVP(1.1).
Without impulse, the PBVP(1.1) has been studied by many authors, see [8–11] and the references therein. In particular, Shen and Li [11] considered the existence of coupled mild -periodic quasisolution pair for the following periodic boundary value problem (PBVP) in
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ5_HTML.gif)
where is continuous. Under one of the following situations:
(i) is a compact semigroup,
(ii) is regular in
and
is continuous in operator norm for
,
they built a mixed monotone iterative method for the PBVP(1.5), and they proved that, if the PBVP(1.5) has coupled lower and upper quasisolutions (i.e., and without impulse in (1.2) and (1.3))
and
with
, nonlinear term
satisfies one of the following conditions:
(F1) is mixed monotone,
(F2)There exists a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ6_HTML.gif)
and is nonincreasing on
.
Then the PBVP(1.5) has minimal and maximal coupled mild -periodic quasisolutions between
and
, which can be obtained by monotone iterative sequences from
and
. But conditions
and
are difficult to satisfy in applications except some special situations.
In this paper, by constructing a mixed monotone iterative technique under a new concept of upper and lower solutions, we will discuss the existence of mild -periodic (
-quasi) solutions for the impulsive evolution Equation(1.1) in an ordered Banach space
. In our results, we will delete conditions
and
for the operator semigroup
, and improve conditions
and
for nonlinearity
. In addition, we only require that the nonlinear term
satisfies weak Carathé odory condition:
(1)for each is strongly measurable.
(2)for a.e. is subcontinuous, namely, there exists
with mes
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ7_HTML.gif)
for any , and
.
Our main result is as follows:
Theorem 1.2.
Let be an ordered and weakly sequentially complete Banach space, whose positive cone
is normal,
be a closed linear operator and
generate a positive
-semigroup
in
. If the PBVP(1.1) has coupled lower and upper
-quasisolutions
and
with
, nonlinear term
and impulsive functions
's satisfy the following conditions
(H1) There exist constants and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ8_HTML.gif)
for any , and
.
(H2) Impulsive function is continuous, and for any
, it satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ9_HTML.gif)
for any , and
.
then the PBVP(1.1) has minimal and maximal coupled mild -periodic
-quasisolutions between
and
, which can be obtained by monotone iterative sequences starting from
and
.
Evidently, condition contains conditions
and
. Hence, even without impulse in PBVP(1.1), Theorem 1.2 still extends the results in [10, 11].
The proof of Theorem 1.2 will be shown in the next section. In Section 2, we also discuss the existence of mild -periodic solutions for the PBVP(1.1) between coupled lower and upper
-quasisolutions (see Theorem 2.3). In Section 3, the results obtained will be applied to a class of partial differential equations of parabolic type.
2. Proof of the Main Results
Let be a Banach space,
be a closed linear operator, and
generate a
-semigroup
in
. Then there exist constants
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ10_HTML.gif)
Definition 2.1.
A -semigroup
is said to be exponentially stable in
if there exist constants
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ11_HTML.gif)
Let . Denote by
the Banach space of all continuous
-value functions on interval
with the norm
. It is well-known ([12, Chapter 4, Theorem
]) that for any
and
, the initial value problem(IVP) of linear evolution equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ12_HTML.gif)
has a unique classical solution expressed by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ13_HTML.gif)
If and
, the function
given by (2.4) belongs to
. We call it a mild solution of the IVP(2.3).
To prove Theorem 1.2, for any , we consider the periodic boundary value problem (PBVP) of linear impulsive evolution equation in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ14_HTML.gif)
where .
Lemma 2.2.
Let be an exponentially stable
-semigroup in
. Then for any
and
, the linear PBVP(2.5) has a unique mild solution
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ15_HTML.gif)
where .
Proof.
For any , we first show that the initial value problem (IVP) of linear impulsive evolution equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ16_HTML.gif)
has a unique mild solution given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ17_HTML.gif)
where and
.
Let . Let
. If
is a mild solution of the linear IVP(2.7), then the restriction of
on
satisfies the initial value problem (IVP) of linear evolution equation without impulse
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ18_HTML.gif)
Hence, on ,
can be expressed by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ19_HTML.gif)
Iterating successively in the above equality with for
, we see that
satisfies (2.8).
Inversely, we can verify directly that the function defined by (2.8) is a solution of the linear IVP(2.7). Hence the linear IVP(2.7) has a unique mild solution
given by (2.8).
Next, we show that the linear PBVP(2.5) has a unique mild solution given by (2.6).
If a function defined by (2.8) is a solution of the linear PBVP(2.5), then
, namely,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ20_HTML.gif)
Since is exponentially stable, we define an equivalent norm in
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ21_HTML.gif)
Then and
, and especially,
. It follows that
has a bounded inverse operator
, which is a positive operator when
is a positive semigroup. Hence we choose
. Then
is the unique initial value of the IVP(2.7) in
, which satisfies
. Combining this fact with (2.8), it follows that (2.6) is satisfied.
Inversely, we can verify directly that the function defined by (2.6) is a solution of the linear PBVP(2.5). Therefore, the conclusion of Lemma 2.2 holds.
Evidently, is also an ordered Banach space with the partial order "
" reduced by positive function 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
. From Lemma 2.2, if
is a positive
-semigroup,
and
, then the mild solution
of the linear PBVP(2.5) satisfies
.
Proof of Theorem 1.2.
We first show that for any
and
. Since
for any
, from the assumption
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ22_HTML.gif)
Namely, . From the normality of cone
in
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ23_HTML.gif)
Combining this fact with the fact that is strongly measurable, it follows that
. Therefore, for any
, we consider the periodic boundary value problem(PBVP) of impulsive evolution equation in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ24_HTML.gif)
where . Let
be large enough such that
(otherwise, replacing
by
, the assumption
still holds). Then
generates an exponentially stable
-semigroup
. Obviously,
is a positive
-semigroup and
for
. From Lemma 2.2, the PBVP(2.15) has a unique mild solution
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ25_HTML.gif)
Let . We define an operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ26_HTML.gif)
Then the coupled mild -periodic
-quasisolution of the PBVP(1.1) is equivalent to the coupled fixed point of operator
.
Next, we will prove that the operator has coupled fixed points on
. For this purpose, we first show that
is a mixed monotone operator and
. In fact, for any
, from assumptions
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ27_HTML.gif)
Since is a positive
-semigroup, it follows that
is a positive operator. Then
. Hence from (2.17) we see that
, which implies that
is a mixed monotone operator. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ28_HTML.gif)
from Lemma 2.2 and (1.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ29_HTML.gif)
for . Especially, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ30_HTML.gif)
Combining this inequality with , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ31_HTML.gif)
On the other hand, from (2.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ32_HTML.gif)
Therefore, for all
. It implies that
. Similarly, we can prove that
.
Now, we define sequences and
by the iterative scheme
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ33_HTML.gif)
Then from the mixed monotonicity of operator , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ34_HTML.gif)
Therefore, for any ,
and
are monotone order-bounded sequences in
. Noticing that
is a weakly sequentially complete Banach space, then
and
are relatively compact in
. Combining this fact with the monotonicity of (2.25) and the normality of cone
in
, it follows that
and
are uniformly convergent in
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ35_HTML.gif)
Then are strongly measurable, and
for any
. Hence,
.
At last, we show that and
are coupled mild
-periodic
-quasisolutions of the PBVP(1.1). For any
, from subcontinuity of
and continuity of
's, there exists
with mes
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ36_HTML.gif)
Hence, for any and
, denote by
the adjoint operator of
, then
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ37_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ38_HTML.gif)
From Lebesgue's dominated convergence theorem, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ39_HTML.gif)
Hence, from (2.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ40_HTML.gif)
On the other hand, it follows from (2.26) that . Hence
. By the uniqueness of limits, we can deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ41_HTML.gif)
By the arbitrariness of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ42_HTML.gif)
Similarly, we can prove that . Therefore,
is coupled mild
-periodic
-quasisolution pair of the PBVP(1.1).
Now, we discuss the existence of mild -periodic solutions for the PBVP(1.1) on
. We assume that the following assumptions are also satisfied:
(H3) there exists a constant with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ43_HTML.gif)
for any , where
,
(H4) there exist positive constants with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ44_HTML.gif)
for any
Then we have the following existence and uniqueness result in general ordered Banach space.
Theorem 2.3.
Let be an ordered Banach space, whose positive cone
is normal,
be a closed linear operator, and
generate a positive
-semigroup
in
. If the PBVP(1.1) has coupled lower and upper
-quasisolution
and
with
, nonlinear term
and impulsive functions
's satisfy the following assumptions:
(H1)* there exist constants and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ45_HTML.gif)
for any , and
.
And , then the PBVP(1.1) has a unique mild
-periodic solution
on
.
Proof.
From the proof of Theorem 1.2, when the conditions and
are satisfied, the iterative sequences
and
defined by (2.24) satisfy (2.25). We show that there exists a unique
such that
. For any
, from
, (2.17), (2.24) and (2.25), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ46_HTML.gif)
By means of the normality of cone in
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ47_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ48_HTML.gif)
by Repeating the using of the above inequality, we can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ49_HTML.gif)
as . Then there exists a unique
such that
. Therefore, let
in (2.24), from the continuity of operator
, we have
, which means that
is a unique mild
-periodic solution of the PBVP(1.1).
3. An Example
Let be a bounded domain with a sufficiently smooth boundary
. Let
, and
,
. Consider the existence of mild solutions for the boundary value problem of parabolic type:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ50_HTML.gif)
where is the Laplace operator,
. Let
equipped with the
-norm
,
. Then
is a generating normal cone in
. Consider the operator
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ51_HTML.gif)
Then generates an analytic semigroup
in
. By the maximum principle of the equations of parabolic type, it is easy to prove that
is a positive
-semigroup in
. Let
be the first eigenvalue of operator
and
be a corresponding positive eigenvector. For solving the problem (3.1), the following assumptions are needed.
(i) There exists a constant such that
(a) ,
,
,
(b) ,
,
,
,
(ii)
(a) The partial derivative of on
is continuous on any bounded domain.
(b) The partial derivative of on
has upper bound, and
.
-
(iii)
For any
with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ52_HTML.gif)
Let and
be defined by
and by
. Then the problem (3.1) can be transformed into the PBVP(1.1). Assumption
implies that
and
are coupled lower and upper
-quasisolutions of the PBVP(1.1). From assumption (ii)(a), there exists a constant
such that, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ53_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ54_HTML.gif)
for any and
. Hence for any
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ55_HTML.gif)
Therefore, for any with
, from the assumption
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F293410/MediaObjects/13660_2009_Article_2112_Equ56_HTML.gif)
That is, assumption is satisfied. From
, it is easy to see that assumption
is satisfied. Therefore, the following result is deduced from Theorem 1.2.
Theorem 3.1.
If the assumptions are satisfied, then the problem (3.1) has coupled mild
-periodic
-quasisolution pair on
.
Remark 3.2.
In applications of partial differential equations, we often choose Banach space as working space, which is weakly sequentially complete. Hence the result in Theorem 1.2 is more valuable in applications. In particular, we obtain a unique mild
-periodic solution of the PBVP(1.1) in general ordered Banach space in Theorem 2.3.
Remark 3.3.
If , then the coupled lower and upper
-quasisolutions are equivalent to coupled lower and upper quasisolutions of the PBVP(1.1). Since condition
contains conditions
and
, even without impulse in PBVP(1.1), the results in this paper still extend the results in [10, 11].
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Acknowledgments
The author is very grateful to the reviewers for their helpful comments and suggestions. Research supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNUKJCXGC-3-47.
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Yang, H. Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces. J Inequal Appl 2010, 293410 (2010). https://doi.org/10.1155/2010/293410
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DOI: https://doi.org/10.1155/2010/293410