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Nonlocal Impulsive Cauchy Problems for Evolution Equations
Advances in Difference Equations volume 2011, Article number: 784161 (2011)
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:

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

,
,
,
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

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

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

In this paper, for , let
and

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:

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

For , we denote by
the set

. 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

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

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

for any ,  
, and

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

(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

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

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

In addition, we introduce the decomposition , where

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

Thus,

which implies that is a contraction by condition (3.6).
Secondly, we prove that ,
,
are completely continuous operators. Let
be a sequence in
with

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

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

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

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

Since is relatively compact in
, we conclude that

which implies that is equicontinuous on
. Therefore,
is a compact operator.
Now, we prove the compactness of . For this purpose, let

Note that

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

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

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

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,

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

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

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

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

For , let
,
,
,
, and we define
by

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

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

Now, for , we have

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
,

that is,

Therefore, is Lipschitz continuous with constant
.
Clearly, is precompact in
, and so is
in
.
Thus, by (3.29) and Lemma 2.4, we obtain

By (3.6), , which implies
. Consequently,
is precompact in
.
Step 2.
is precompact in
.
For , let

and define by

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

So, for ,
, we have

where

According to the proof of Step 1, we know that

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

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

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

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

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

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

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

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

as . Thus,

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

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

Letting in both sides, we obtain

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

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:

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

with domain

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
,

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

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

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 .
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Acknowledgments
The authors would like to thank the referees for helpful comments and suggestions. J. Liang acknowledges support from the NSF of China (10771202) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805). Z. Fan acknowledges support from the NSF of China (11001034) and the Research Fund for Shanghai Postdoctoral Scientific Program (10R21413700).
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Liang, J., Fan, Z. Nonlocal Impulsive Cauchy Problems for Evolution Equations. Adv Differ Equ 2011, 784161 (2011). https://doi.org/10.1155/2011/784161
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DOI: https://doi.org/10.1155/2011/784161