- Research Article
- Open Access

# Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces

- He Yang
^{1}Email author

**2010**:293410

https://doi.org/10.1155/2010/293410

© He Yang. 2010

**Received: **29 December 2009

**Accepted: **3 September 2010

**Published: **14 September 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.

## Keywords

- Mild Solution
- Initial Value Problem
- Lower Solution
- Impulsive Differential Equation
- Order Banach Space

## 1. Introduction and Main Result

in an ordered Banach space
, where
is a closed linear operator and
generates a
-semigroup
in
;
only satisfies weak Carath*é* 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).

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.

where and for any , is an identity operator. If , then is called a mild -periodic solution of the PBVP(1.1).

where is continuous. Under one of the following situations:

(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:

(*F2*)There exists a constant
such that

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 Carath*é* odory condition:

(1)for each is strongly measurable.

(2)for a.e. is subcontinuous, namely, there exists with mes such that

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

(*H2*) Impulsive function
is continuous, and for any
, it satisfies

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

Definition 2.1.

If and , the function given by (2.4) belongs to . We call it a mild solution of the IVP(2.3).

Lemma 2.2.

Proof.

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).

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.

Then the coupled mild -periodic -quasisolution of the PBVP(1.1) is equivalent to the coupled fixed point of operator .

Therefore, for all . It implies that . Similarly, we can prove that .

Then are strongly measurable, and for any . Hence, .

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:

(*H4*) there exist positive constants
with
such that

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

And , then the PBVP(1.1) has a unique mild -periodic solution on .

Proof.

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

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

(ii)

(a) The partial derivative of on is continuous on any bounded domain.

(b) The partial derivative of on has upper bound, and .

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].

## Declarations

### 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.

## Authors’ Affiliations

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