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

Open Access
Research Article

Abstract

By constructing a mixed monotone iterative technique under a new concept of upper and lower solutions, some existence theorems of mild Open image in new window -periodic ( Open image in new window -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

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, 2, 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, 5, 6, 7] and the references therein. In this paper, by using a mixed monotone iterative technique in the presence of coupled lower and upper Open image in new window -quasisolutions, we consider the existence of mild Open image in new window -periodic ( Open image in new window -quasi)solutions for the periodic boundary value problem (PBVP) of impulsive evolution equations

in an ordered Banach space Open image in new window , where Open image in new window is a closed linear operator and Open image in new window generates a Open image in new window -semigroup Open image in new window in Open image in new window ; Open image in new window only satisfies weak Carathé odory condition, Open image in new window , Open image in new window is a constant; Open image in new window ; Open image in new window is an impulsive function, Open image in new window ; Open image in new window denotes the jump of Open image in new window at Open image in new window , that is, Open image in new window , where Open image in new window and Open image in new window represent the right and left limits of Open image in new window at Open image in new window , respectively. Let Open image in new window is continuous at Open image in new window and left continuous at Open image in new window , and Open image in new window exists, Open image in new window . Evidently, Open image in new window is a Banach space with the norm Open image in new window . Let Open image in new window , Open image in new window . Denote by Open image in new window the Banach space generated by Open image in new window with the norm Open image in new window . An abstract function Open image in new window is called a solution of the PBVP(1.1) if Open image in new window satisfies all the equalities of (1.1).

Let Open image in new window be an ordered Banach space with the norm Open image in new window and the partial order " Open image in new window ", whose positive cone Open image in new window is normal with a normal constant Open image in new window . Let Open image in new window . If functions Open image in new window satisfy

we call Open image in new window coupled lower and upper Open image in new window -quasisolutions of the PBVP(1.1). Only choosing " Open image in new window " in (1.2) and (1.3), we call Open image in new window coupled Open image in new window -periodic Open image in new window -quasisolution pair of the PBVP(1.1). Furthermore, if Open image in new window , we call Open image in new window an Open image in new window -periodic solution of the PBVP(1.1).

Definition 1.1.

Abstract functions Open image in new window are called a coupled mild Open image in new window -periodic Open image in new window -quasisolution pair of the PBVP(1.1) if Open image in new window and Open image in new window satisfy the following integral equations:

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

Without impulse, the PBVP(1.1) has been studied by many authors, see [8, 9, 10, 11] and the references therein. In particular, Shen and Li [11] considered the existence of coupled mild Open image in new window -periodic quasisolution pair for the following periodic boundary value problem (PBVP) in Open image in new window :

where Open image in new window is continuous. Under one of the following situations:

(i) Open image in new window is a compact semigroup,

(ii) Open image in new window is regular in Open image in new window and Open image in new window is continuous in operator norm for Open image in new window ,

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., Open image in new window and without impulse in (1.2) and (1.3)) Open image in new window and Open image in new window with Open image in new window , nonlinear term Open image in new window satisfies one of the following conditions:

(F1) Open image in new window is mixed monotone,

(F2)There exists a constant Open image in new window such that

and Open image in new window is nonincreasing on Open image in new window .

Then the PBVP(1.5) has minimal and maximal coupled mild Open image in new window -periodic quasisolutions between Open image in new window and Open image in new window , which can be obtained by monotone iterative sequences from Open image in new window and Open image in new window . But conditions Open image in new window and Open image in new window 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 Open image in new window -periodic ( Open image in new window -quasi) solutions for the impulsive evolution Equation(1.1) in an ordered Banach space Open image in new window . In our results, we will delete conditions Open image in new window and Open image in new window for the operator semigroup Open image in new window , and improve conditions Open image in new window and Open image in new window for nonlinearity Open image in new window . In addition, we only require that the nonlinear term Open image in new window satisfies weak Carathé odory condition:

(1)for each Open image in new window is strongly measurable.

(2)for a.e. Open image in new window is subcontinuous, namely, there exists Open image in new window with mes Open image in new window such that

for any Open image in new window , and Open image in new window .

Our main result is as follows:

Theorem 1.2.

Let Open image in new window be an ordered and weakly sequentially complete Banach space, whose positive cone Open image in new window is normal, Open image in new window be a closed linear operator and Open image in new window generate a positive Open image in new window -semigroup Open image in new window in Open image in new window . If the PBVP(1.1) has coupled lower and upper Open image in new window -quasisolutions Open image in new window and Open image in new window with Open image in new window , nonlinear term Open image in new window and impulsive functions Open image in new window 's satisfy the following conditions

(H1) There exist constants Open image in new window and Open image in new window such that

for any Open image in new window , and Open image in new window .

(H2) Impulsive function Open image in new window is continuous, and for any Open image in new window , it satisfies

for any Open image in new window , and Open image in new window .

then the PBVP(1.1) has minimal and maximal coupled mild Open image in new window -periodic Open image in new window -quasisolutions between Open image in new window and Open image in new window , which can be obtained by monotone iterative sequences starting from Open image in new window and Open image in new window .

Evidently, condition Open image in new window contains conditions Open image in new window and Open image in new window . 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 Open image in new window -periodic solutions for the PBVP(1.1) between coupled lower and upper Open image in new window -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.

A Open image in new window -semigroup Open image in new window is said to be exponentially stable in Open image in new window if there exist constants Open image in new window and Open image in new window such that
Let Open image in new window . Denote by Open image in new window the Banach space of all continuous Open image in new window -value functions on interval Open image in new window with the norm Open image in new window . It is well-known ([12, Chapter 4, Theorem Open image in new window ]) that for any Open image in new window and Open image in new window , the initial value problem(IVP) of linear evolution equation
has a unique classical solution Open image in new window expressed by

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

To prove Theorem 1.2, for any Open image in new window , we consider the periodic boundary value problem (PBVP) of linear impulsive evolution equation in Open image in new window

where Open image in new window .

Lemma 2.2.

Let Open image in new window be an exponentially stable Open image in new window -semigroup in Open image in new window . Then for any Open image in new window and Open image in new window , the linear PBVP(2.5) has a unique mild solution Open image in new window given by

where Open image in new window .

Proof.

For any Open image in new window , we first show that the initial value problem (IVP) of linear impulsive evolution equation
has a unique mild solution Open image in new window given by

where Open image in new window and Open image in new window .

Let Open image in new window . Let Open image in new window . If Open image in new window is a mild solution of the linear IVP(2.7), then the restriction of Open image in new window on Open image in new window satisfies the initial value problem (IVP) of linear evolution equation without impulse

Iterating successively in the above equality with Open image in new window for Open image in new window , we see that Open image in new window satisfies (2.8).

Inversely, we can verify directly that the function Open image in new window defined by (2.8) is a solution of the linear IVP(2.7). Hence the linear IVP(2.7) has a unique mild solution Open image in new window given by (2.8).

Next, we show that the linear PBVP(2.5) has a unique mild solution Open image in new window given by (2.6).

If a function Open image in new window defined by (2.8) is a solution of the linear PBVP(2.5), then Open image in new window , namely,
Since Open image in new window is exponentially stable, we define an equivalent norm in Open image in new window by

Then Open image in new window and Open image in new window , and especially, Open image in new window . It follows that Open image in new window has a bounded inverse operator Open image in new window , which is a positive operator when Open image in new window is a positive semigroup. Hence we choose Open image in new window . Then Open image in new window is the unique initial value of the IVP(2.7) in Open image in new window , which satisfies Open image in new window . Combining this fact with (2.8), it follows that (2.6) is satisfied.

Inversely, we can verify directly that the function Open image in new window defined by (2.6) is a solution of the linear PBVP(2.5). Therefore, the conclusion of Lemma 2.2 holds.

Evidently, Open image in new window is also an ordered Banach space with the partial order " Open image in new window " reduced by positive function cone Open image in new window . Open image in new window is also normal with the same normal constant Open image in new window . For Open image in new window with Open image in new window , we use Open image in new window to denote the order interval Open image in new window in Open image in new window , and Open image in new window to denote the order interval Open image in new window in Open image in new window . From Lemma 2.2, if Open image in new window is a positive Open image in new window -semigroup, Open image in new window and Open image in new window , then the mild solution Open image in new window of the linear PBVP(2.5) satisfies Open image in new window .

Proof of Theorem 1.2.

Combining this fact with the fact that Open image in new window is strongly measurable, it follows that Open image in new window . Therefore, for any Open image in new window , we consider the periodic boundary value problem(PBVP) of impulsive evolution equation in Open image in new window
where Open image in new window . Let Open image in new window be large enough such that Open image in new window (otherwise, replacing Open image in new window by Open image in new window , the assumption Open image in new window still holds). Then Open image in new window generates an exponentially stable Open image in new window -semigroup Open image in new window . Obviously, Open image in new window is a positive Open image in new window -semigroup and Open image in new window for Open image in new window . From Lemma 2.2, the PBVP(2.15) has a unique mild solution Open image in new window given by

Then the coupled mild Open image in new window -periodic Open image in new window -quasisolution of the PBVP(1.1) is equivalent to the coupled fixed point of operator Open image in new window .

Next, we will prove that the operator Open image in new window has coupled fixed points on Open image in new window . For this purpose, we first show that Open image in new window is a mixed monotone operator and Open image in new window . In fact, for any Open image in new window , from assumptions Open image in new window and Open image in new window , we have
Since Open image in new window is a positive Open image in new window -semigroup, it follows that Open image in new window is a positive operator. Then Open image in new window . Hence from (2.17) we see that Open image in new window , which implies that Open image in new window is a mixed monotone operator. Since
from Lemma 2.2 and (1.2), we have
for Open image in new window . Especially, we have
Combining this inequality with Open image in new window , it follows that
On the other hand, from (2.17), we have

Therefore, Open image in new window for all Open image in new window . It implies that Open image in new window . Similarly, we can prove that Open image in new window .

Now, we define sequences Open image in new window and Open image in new window by the iterative scheme
Then from the mixed monotonicity of operator Open image in new window , we have
Therefore, for any Open image in new window , Open image in new window and Open image in new window are monotone order-bounded sequences in Open image in new window . Noticing that Open image in new window is a weakly sequentially complete Banach space, then Open image in new window and Open image in new window are relatively compact in Open image in new window . Combining this fact with the monotonicity of (2.25) and the normality of cone Open image in new window in Open image in new window , it follows that Open image in new window and Open image in new window are uniformly convergent in Open image in new window . Let

Then Open image in new window are strongly measurable, and Open image in new window for any Open image in new window . Hence, Open image in new window .

At last, we show that Open image in new window and Open image in new window are coupled mild Open image in new window -periodic Open image in new window -quasisolutions of the PBVP(1.1). For any Open image in new window , from subcontinuity of Open image in new window and continuity of Open image in new window 's, there exists Open image in new window with mes Open image in new window such that
On the other hand, we have
From Lebesgue's dominated convergence theorem, we have
Hence, from (2.17), we have
On the other hand, it follows from (2.26) that Open image in new window . Hence Open image in new window . By the uniqueness of limits, we can deduce that
By the arbitrariness of Open image in new window , we have

Similarly, we can prove that Open image in new window . Therefore, Open image in new window is coupled mild Open image in new window -periodic Open image in new window -quasisolution pair of the PBVP(1.1).

Now, we discuss the existence of mild Open image in new window -periodic solutions for the PBVP(1.1) on Open image in new window . We assume that the following assumptions are also satisfied:

(H3) there exists a constant Open image in new window with Open image in new window such that

for any Open image in new window , where Open image in new window ,

(H4) there exist positive constants Open image in new window with Open image in new window such that

for any Open image in new window

Then we have the following existence and uniqueness result in general ordered Banach space.

Theorem 2.3.

Let Open image in new window be an ordered Banach space, whose positive cone Open image in new window is normal, Open image in new window be a closed linear operator, and Open image in new window generate a positive Open image in new window -semigroup Open image in new window in Open image in new window . If the PBVP(1.1) has coupled lower and upper Open image in new window -quasisolution Open image in new window and Open image in new window with Open image in new window , nonlinear term Open image in new window and impulsive functions Open image in new window 's satisfy the following assumptions:

(H1)* there exist constants Open image in new window and Open image in new window such that

for any Open image in new window , and Open image in new window .

And Open image in new window , then the PBVP(1.1) has a unique mild Open image in new window -periodic solution Open image in new window on Open image in new window .

Proof.

From the proof of Theorem 1.2, when the conditions Open image in new window and Open image in new window are satisfied, the iterative sequences Open image in new window and Open image in new window defined by (2.24) satisfy (2.25). We show that there exists a unique Open image in new window such that Open image in new window . For any Open image in new window , from Open image in new window , (2.17), (2.24) and (2.25), we have
By means of the normality of cone Open image in new window in Open image in new window , we have
Therefore
by Repeating the using of the above inequality, we can obtain that

as Open image in new window . Then there exists a unique Open image in new window such that Open image in new window . Therefore, let Open image in new window in (2.24), from the continuity of operator Open image in new window , we have Open image in new window , which means that Open image in new window is a unique mild Open image in new window -periodic solution of the PBVP(1.1).

3. An Example

Let Open image in new window be a bounded domain with a sufficiently smooth boundary Open image in new window . Let Open image in new window , and Open image in new window , Open image in new window . Consider the existence of mild solutions for the boundary value problem of parabolic type:

Then Open image in new window generates an analytic semigroup Open image in new window in Open image in new window . By the maximum principle of the equations of parabolic type, it is easy to prove that Open image in new window is a positive Open image in new window -semigroup in Open image in new window . Let Open image in new window be the first eigenvalue of operator Open image in new window and Open image in new window be a corresponding positive eigenvector. For solving the problem (3.1), the following assumptions are needed.

(i) There exists a constant Open image in new window such that

(a) Open image in new window , Open image in new window , Open image in new window , Open image in new window

(b) Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window

(ii)

(a) The partial derivative of Open image in new window on Open image in new window is continuous on any bounded domain.

(b) The partial derivative of Open image in new window on Open image in new window has upper bound, and Open image in new window .

  1. (iii)
     
Let Open image in new window and Open image in new window be defined by Open image in new window and by Open image in new window . Then the problem (3.1) can be transformed into the PBVP(1.1). Assumption Open image in new window implies that Open image in new window and Open image in new window are coupled lower and upper Open image in new window -quasisolutions of the PBVP(1.1). From assumption (ii)(a), there exists a constant Open image in new window such that, for any Open image in new window , we have
This implies that
Therefore, for any Open image in new window with Open image in new window , from the assumption Open image in new window , we have

That is, assumption Open image in new window is satisfied. From Open image in new window , it is easy to see that assumption Open image in new window is satisfied. Therefore, the following result is deduced from Theorem 1.2.

Theorem 3.1.

If the assumptions Open image in new window are satisfied, then the problem (3.1) has coupled mild Open image in new window -periodic Open image in new window -quasisolution pair on Open image in new window .

Remark 3.2.

In applications of partial differential equations, we often choose Banach space Open image in new window 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 Open image in new window -periodic solution of the PBVP(1.1) in general ordered Banach space in Theorem 2.3.

Remark 3.3.

If Open image in new window , then the coupled lower and upper Open image in new window -quasisolutions are equivalent to coupled lower and upper quasisolutions of the PBVP(1.1). Since condition Open image in new window contains conditions Open image in new window and Open image in new window , even without impulse in PBVP(1.1), the results in this paper still extend the results in [10, 11].

Notes

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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© He Yang. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouChina

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