Nonlocal Impulsive Cauchy Problems for Evolution Equations

Abstract

Of concern is the existence of solutions to nonlocal impulsive Cauchy problems for evolution equations. Combining the techniques of operator semigroups, approximate solutions, noncompact measures and the fixed point theory, new existence theorems are obtained, which generalize and improve some previous results since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required. An application to partial differential equations is also presented.

1. Introduction

Impulsive equations arise from many different real processes and phenomena which appeared in physics, chemical technology, population dynamics, biotechnology, medicine, and economics. They have in recent years been an object of investigations with increasing interest. For more information on this subject, see for instance, the papers (cf., e.g., [1–6]) and references therein.

On the other hand, Cauchy problems with nonlocal conditions are appropriate models for describing a lot of natural phenomena, which cannot be described using classical Cauchy problems. That is why in recent years they have been studied by many researchers (cf., e.g., [4, 7–12] and references therein).

In [4], the authors combined the two directions and studied firstly a class of nonlocal impulsive Cauchy problems for evolution equations by investigating the existence for mild (in generalized sense) solutions to the problems. In this paper, we study further the existence of solutions to the following nonlocal impulsive Cauchy problem for evolution equations:

(1.1)

where is the infinitesimal generator of an analytic semigroup and is a real Banach space endowed with the norm ,

(1.2)

, , , are given continuous functions to be specified later.

By going a new way, that is, by combining operator semigroups, the techniques of approximate solutions, noncompact measures, and the fixed point theory, we obtain new existence results for problem (1.1), which generalize and improve some previous theorems since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required in the present paper.

The organization of this work is as follows. In Section 2, we recall some definitions, and facts about fractional powers of operators, mild solutions and Hausdorff measure of noncompactness. In Section 3, we give the existence results for problem (1.1) when the nonlocal item and impulsive functions are only assumed to be continuous. In Section 4, we give an example to illustrate our abstract results.

2. Preliminaries

Let be a real Banach space. We denote by the space of -valued continuous functions on with the norm

(2.1)

and by the space of -valued Bochner integrable functions on with the norm . Let

(2.2)

It is easy to check that is a Banach space with the norm

(2.3)

In this paper, for , let and

(2.4)

Throughout this paper, we assume the following.

(H1) The operator is the infinitesimal generator of a compact analytic semigroup on Banach space and (the resolvent set of ).

In the remainder of this work, .

Under the above conditions, it is possible to define the fractional power , , of as closed linear operators. And it is known that the following properties hold.

Theorem 2.1 (see [13, Pages 69–75]).

Let and assume that (H1) holds. Then,

• (1) is a Banach space with the norm for ,

• (2) for ,

• (3) for and ,

• (4) for every , is bounded on and there exists such that

  

  (2.5)

• (5) is a bounded linear operator in with ,

• (6) if , then .

We denote by that the Banach space endowed the graph norm from now on.

Definition 2.2.

A function is said to be a mild solution of (1.1) on if the function is integrable on for all and the following integral equation is satisfied:

(2.6)

To discuss the compactness of subsets of , we let , ,

(2.7)

For , we denote by the set

(2.8)

. Then it is easy to see that the following result holds.

Lemma 2.3.

A set is precompact in if and only if the set is precompact in for every .

Next, we recall that the Hausdorff measure of noncompactness on each bounded subset of Banach space is defined by

(2.9)

Some basic properties of are given in the following Lemma.

Lemma 2.4 (see [14]).

Let be a real Banach space and let be bounded. Then,

• (1) is precompact if and only if ;

• (2) , where and mean the closure and convex hull of , respectively;

• (3) when ;

• (4) , where ;

• (5) ;

• (6) for any ;

• (7) let be a Banach space and Lipschitz continuous with constant . Then for all being bounded.

We note that a continuous map is an -contraction if there exists a positive constant such that for all bounded closed .

Lemma 2.5 (see Darbo-Sadovskii's fixed point theorem in [14]).

If is bounded closed and convex, and is an -contraction, then the map has at least one fixed point in .

3. Main Results

In this section, by using the techniques of approximate solutions and fixed points, we establish a result on the existence of mild solutions for the nonlocal impulsive problem (1.1) when the nonlocal item and the impulsive functions are only assumed to be continuous in and , respectively.

In practical applications, the values of for near zero often do not affect . For example, it is the case when

(3.1)

So, to prove our main results, we introduce the following assumptions.

(H2) is a continuous function, and there is a such that for any with , . Moreover, there exist such that for any .

(H3)There exists a such that is a continuous function, and for any with , . Moreover, there exist such that

(3.2)

for any ,  , and

(3.3)

for any , .

(H4)The function is continuous a.e. ; the function is strongly measurable for all . Moreover, for each , there exists a function such that for a.e. and all , and

(3.4)

(H5) is continuous for every , and there exist positive numbers such that for any and .

We note that, by Theorem 2.1, there exist and such that and

(3.5)

For simplicity, in the following we set and will substitute by below.

Theorem 3.1.

Let (H1)–(H5) hold. Then the nonlocal impulsive Cauchy problem (1.1) has at least one mild solution on , provided

(3.6)

To prove the theorem, we need some lemmas. Next, for , we denote by the maps defined by

(3.7)

In addition, we introduce the decomposition , where

(3.8)

for and .

Lemma 3.2.

Assume that all the conditions in Theorem 3.1 are satisfied. Then for any , the map defined by (3.7) has at least one fixed point .

Proof.

To prove the existence of a fixed point for , we will use Darbu-Sadovskii's fixed point theorem.

Firstly, we prove that the map is a contraction on . For this purpose, let . Then for each and by condition (H3), we have

(3.9)

Thus,

(3.10)

which implies that is a contraction by condition (3.6).

Secondly, we prove that , , are completely continuous operators. Let be a sequence in with

(3.11)

in . By the continuity of with respect to the second argument, we deduce that for each , converges to in , and we have

(3.12)

Then by the continuity of , , , and using the dominated convergence theorem, we get

(3.13)

in , which implies that are continuous on .

Next, for the compactness of we refer to the proof of [4, Theorem  3.1].

For and any bounded subset of , we have

(3.14)

which implies that is relatively compact in for every by the compactness of . On the other hand, for , we have

(3.15)

Since is relatively compact in , we conclude that

(3.16)

which implies that is equicontinuous on . Therefore, is a compact operator.

Now, we prove the compactness of . For this purpose, let

(3.17)

Note that

(3.18)

Thus according to Lemma 2.3, we only need to prove that

(3.19)

is precompact in , as the remaining cases for , , can be dealt with in the same way; here is any bounded subset in . And, we recall that , , which means that

(3.20)

Thus, by the compactness of , we know that is relatively compact in for every .

Next, for , we have

(3.21)

Thus, the set is equicontinuous due to the compactness of and the strong continuity of operator . By the Arzela-Ascoli theorem, we conclude that is precompact in . The same idea can be used to prove that is precompact for each . Therefore, is precompact in , that is, the operator is compact.

Thus, for any bounded subset , we have by Lemma 2.4,

(3.22)

Hence, the map is an -contraction in .

Now, in order to apply Lemma 2.5, it remains to prove that there exists a constant such that . Suppose this is not true; then for each positive integer , there are and such that . Then

(3.23)

Dividing on both sides by and taking the lower limit as , we obtain that

(3.24)

This is a contradiction with inequality (3.6). Therefore, there exists such that the mapping maps into itself. By Darbu-Sadovskii's fixed point theorem, the operator has at least one fixed point in . This completes the proof.

Lemma 3.3.

Assume that all the conditions in Theorem 3.1 are satisfied. Then the set is precompact in for all , where

(3.25)

and is the constant in (H2).

Proof.

The proof will be given in several steps. In the following is a number in .

Step 1.

is precompact in .

For , define by

(3.26)

For , let , , , , and we define by

(3.27)

By condition (H3), is well defined and for , we have

(3.28)

On the other hand, for , , we have , . So,

(3.29)

Now, for , we have

(3.30)

By the compactness of , , we get that is relatively compact in for every and is equicontinuous on , which implies that is precompact in .

By the same reasoning, is precompact in .

For , we claim that is Lipschitz continuous with constant . In fact, (H3) implies that for every and ,

(3.31)

that is,

(3.32)

Therefore, is Lipschitz continuous with constant .

Clearly, is precompact in , and so is in .

Thus, by (3.29) and Lemma 2.4, we obtain

(3.33)

By (3.6), , which implies . Consequently, is precompact in .

Step 2.

is precompact in .

For , let

(3.34)

and define by

(3.35)

By (H3), is well defined and for , we have

(3.36)

So, for , , we have

(3.37)

where

(3.38)

According to the proof of Step 1, we know that

(3.39)

are all precompact in and is Lipschitz continuous with constant .

Next, we will show that is precompact in . Firstly, it is easy to see that is precompact in . Thus according to Lemma 2.3, it remains to prove that

(3.40)

is precompact in . And, we recall that , , which means that

(3.41)

By Step 1, is precompact in . Without loss of generality, we may suppose that

(3.42)

Therefore, , as in . Thus, by the continuity of and , we get

(3.43)

as , which implies that is relatively compact in . And, for , by the compactness of , , is also relatively compact in . Therefore, is relatively compact in for every .

Next, for , we have

(3.44)

Thus, the set is equicontinuous on due to the compactness of and the strong continuity of operator , . By the Arzela-Ascoli theorem, we conclude that is precompact in . Therefore, is precompact in .

Thus, by Lemma 2.4, we obtain

(3.45)

By (3.6), , which implies . Consequently, is precompact in .

Step 3.

The same idea can be used to prove the compactness of in for , where . This completes the proof.

Proof of Theorem 3.1.

For , , let

(3.46)

where comes from the condition (H2). Then, by condition (H2), .

By Lemma 3.3, without loss of generality, we may suppose that , as . Thus, by the continuity of and , we get

(3.47)

as . Thus,

(3.48)

is precompact in . Moreover, and are both precompact in . And is Lipschitz continuous with constant . Note that

(3.49)

Therefore, by Lemma 2.4, we know that the set is precompact in . Without loss of generality, we may suppose that in . On the other hand, we also have

(3.50)

Letting in both sides, we obtain

(3.51)

which implies that is a mild solution of the nonlocal impulsive problem (1.1). This completes the proof.

Remark 3.4.

From Lemma 3.3 and the above proof, it is easy to see that we can also prove Theorem 3.1 by showing that is precompact in .

The following results are immediate consequences of Theorem 3.5.

Theorem 3.5.

Assume (H1), (H3)–(H5) hold. If , then the impulsive Cauchy problem (1.1) has at least one mild solution on , provided

(3.52)

Theorem 3.6.

Assume (H1), (H2), (H4), and (H5) hold. If , then the nonlocal impulsive problem (1.1) has at least one mild solution on , provided .

Theorem 3.7.

Assume (H1), (H4), and (H5) hold. If , then the impulsive problem (1.1) has at least one mild solution on , provided .

Remark 3.8.

Theorems  3.5-3.6 are new even for many special cases discussed before, since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required.

4. Application

In this section, to illustrate our abstract result, we consider the following differential system:

(4.1)

where , are given real numbers for , , and and are functions to be specified below.

To treat the above system, we take with the norm and we consider the operator defined by

(4.2)

with domain

(4.3)

The operator is the infinitesimal generator of an analytic compact semigroup on . Moreover, has a discrete spectrum, the eigenvalues are , , with the corresponding normalized eigenvectors , and the following properties are satisfied.

• (a) If , then .

• (b) For each , . Moreover, for all .

• (c) For each , . In particular, .

• (d) is given by with the domain .

Assume the following.

• (1) The function is continuously differential with for , , and there exists a real number such that for , . Moreover,

  

  (4.4)

• (2) For each , is continuous, and for each , is measurable and, there exists a function such that for a.e. and all .

• (3) is a continuous function for each , and there exist positive numbers such that for any and .

Define and , respectively, as follows. For ,

(4.5)

From the definition of and assumption (1), it follows that

(4.6)

Thus, system (4.1) can be transformed into the abstract problem (1.1), and conditions (H2), (H3), (H4), and (H5) are satisfied with

(4.7)

If (3.6) holds (it holds when the related constants are small), then according to Theorem 3.1, the problem (4.1) has at least one mild solution in .