Nonlocal Impulsive Cauchy Problems for Evolution Equations
© J. Liang and Z. Fan. 2011
Received: 17 October 2010
Accepted: 19 November 2010
Published: 25 November 2010
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.
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).
, , , 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.
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,
is a Banach space with the norm for ,
for and ,
for every , is bounded on and there exists such that(2.5)
is a bounded linear operator in with ,
if , then .
We denote by that the Banach space endowed the graph norm from now on.
. Then it is easy to see that the following result holds.
A set is precompact in if and only if the set is precompact in for every .
Some basic properties of are given in the following Lemma.
Lemma 2.4 (see ).
Let be a real Banach space and let be bounded. Then,
is precompact if and only if ;
, where and mean the closure and convex hull of , respectively;
, where ;
for any ;
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 ).
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.
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 .
for any , .
(H5) is continuous for every , and there exist positive numbers such that for any and .
For simplicity, in the following we set and will substitute by below.
for and .
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 .
To prove the existence of a fixed point for , we will use Darbu-Sadovskii's fixed point theorem.
which implies that is a contraction by condition (3.6).
in , which implies that are continuous on .
Next, for the compactness of we refer to the proof of [4, Theorem 3.1].
which implies that is equicontinuous on . Therefore, is a compact operator.
Thus, by the compactness of , we know that is relatively compact in for every .
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.
Hence, the map is an -contraction in .
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.
and is the constant in (H2).
The proof will be given in several steps. In the following is a number in .
is precompact in .
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 .
Therefore, is Lipschitz continuous with constant .
Clearly, is precompact in , and so is in .
By (3.6), , which implies . Consequently, is precompact in .
is precompact in .
are all precompact in and is Lipschitz continuous with constant .
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 .
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 .
By (3.6), , which implies . Consequently, is precompact in .
The same idea can be used to prove the compactness of in for , where . This completes the proof.
Proof of Theorem 3.1.
where comes from the condition (H2). Then, by condition (H2), .
which implies that is a mild solution of the nonlocal impulsive problem (1.1). This completes the proof.
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.
Assume (H1), (H2), (H4), and (H5) hold. If , then the nonlocal impulsive problem (1.1) has at least one mild solution on , provided .
Assume (H1), (H4), and (H5) hold. If , then the impulsive problem (1.1) has at least one mild solution on , provided .
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.
where , are given real numbers for , , and and are functions to be specified below.
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.
If , then .
For each , . Moreover, for all .
For each , . In particular, .
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)
For each , is continuous, and for each , is measurable and, there exists a function such that for a.e. and all .
is a continuous function for each , and there exist positive numbers such that for any and .
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 .
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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