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Abstract Semilinear Evolution Equations with Convex-Power Condensing Operators
By using the techniques of convex-power condensing operators and fixed point theorems, we investigate the existence of mild solutions to nonlocal impulsive semilinear differential equations. Two examples are also given to illustrate our main results.
This paper is concerned with the existence of mild solutions for the following impulsive semilinear differential equations with nonlocal conditions where is the infinitesimal generator of strongly continuous semigroup for in a real Banach space and constitutes an impulsive condition. and are -valued functions to be given later.
As far as we know, the first paper dealing with abstract nonlocal initial value problems for semilinear differential equations is due to . Because nonlocal conditions have better effect in the applications than the classical initial ones, many authors have studied the following type of semilinear differential equations under various conditions on ,, and :
For instance, Byszewski and Lakshmikantham  proved the existence and uniqueness of mild solutions for nonlocal semilinear differential equations when and satisfy Lipschitz type conditions. In , Ntouyas and Tsamatos studied the case with compactness conditions. Byszewski and Akca  established the existence of solution to functionaldifferential equation when the semigroup is compact and is convex and compact on a given ball. Subsequently, Benchohra and Ntouyas  discussed the second-order differential equation under compact conditions. The fully nonlinear case was considered by Aizicovici and McKibben , Aizicovici and Lee , Aizicovici and Staicu , García-Falset , Paicu and Vrabie , Obukhovski and Zecca , and Xue [12, 13].
Recently, the theory of impulsive differential inclusions has become an important object of investigation because of its wide applicability in biology, medicine, mechanics, and control theory and in more and more fields. Cardinali and Rubbioni  studied the multivalued impulsive semilinear differential equation by means of the Hausdorff measure of noncompactness. Liang et al.  investigated the nonlocal impulsive problems under the assumptions that is compact, Lipschitz, and is not compact and not Lipschitz, respectively. All these studies are motivated by the practical interests of nonlocal impulsive Cauchy problems. For a more detailed bibliography and exposition on this subject, we refer to [14–18].
The present paper is motivated by the following facts. Firstly, the approach used in [9, 12, 13, 19, 20] relies on the assumption that the coefficient of the function about the measure of noncompactness satisfies a strong inequality, which is difficult to be verified in applications. Secondly, in , it seems that authors have considered the inequality restriction on coefficient function of may be relaxed for impulsive nonlocal differential equations. However, in fact, they only solve the classical initial value problems rather than the nonlocal initial problems . For more details, one can refer to the proof of Theorem 3.1 in  (see the inequalities and in page 5 and the estimations of the measure of noncompactness in page 6 and page 7 of ).
Therefore, we will continue to discuss the impulsive nonlocal differential equations under more general assumptions. Throughout this work, we mainly use the property of convex-power condensing operators and fixed point theorems to obtain the main result (Theorem 10). Indeed, the fixed point theorem about the convex-power condensing operators is an extension for Darbo-Sadovskii’s fixed point theorem. But the former seems more effective than the latter at times for some problems. For example, in  we ever applied the former to study the nonlocal Cauchy problem and obtained more general and interesting existence results. Based on the results obtained, we discuss the impulsive nonlocal differential equations. Fortunately, applying the techniques of convex-power condensing operators and fixed point theorems solves the difficulty involved by coefficient restriction that is, the constraint condition for the coefficient function of is unnecessary (see Theorem 10). Therefore, our results generalize and improve many previous ones in this field, such as [9, 12, 13, 19, 20].
The outline of this paper is as follows. In Section 2, we recall some concepts and facts about the measure of noncompactness, fixed point theorems, and impulsive semilinear differential equations. In Section 3, we obtain the existence results of (1) when is compact in . In Section 4, we discuss the existence result of (1) when is Lipschitz continuous, while Section 5 contains two illustrating examples.
Let be a real Banach space, we introduce the Hausdorff measure of noncompactness defined on each bounded subset of by
Now we recall some basic properties of the Hausdorff measure of noncompactness.
Lemma 1. For all bounded subsets ,, and of , the following properties are satisfied:(1) is precompact if and only if ;(2), where and mean the closure and convex hull of , respectively;(3) when ;(4); (5), for any ;(6), where ;(7)if is a decreasing sequence of nonempty bounded closed subsets of and , then is nonempty and compact in .
The map is said to be -condensing if for every bounded and not relatively compact , we have (see ).
Lemma 2 (see : Darbo-Sadovskii). If is bounded closed and convex, the continuous map is -condensing, then the map has at least one fixed point in .
In the sequel, we will continue to generalize the definition of condensing operator. First of all, we give some notations.
Let be bounded closed and convex, the map , and for every , set where means the closure of the convex hull.
Now we give the definition of a kind of new operator.
Definition 3. Let be bounded closed and convex, the map is said to be -convex-power condensing if there exist , and for every bounded and not relatively compact , we have From this definition, if , one obtains as relatively compact.
Subsequently, we give the fixed point theorem about the convex-power condensing operator.
Lemma 4 (see ). If is bounded closed and convex, the continuous map is -convex-power condensing, then the map has at least one fixed point in .
Throughout this paper, let be a real Banach space. We denote by the Banach space of all continuous functions from to with the norm sup and by the Banach space of all -valued Bochner integrable functions defined on with the norm . Let is a function from intosuch that is continuous atand the left continuous at and the right limitexists for. It is easy to check that is a Banach space with the norm and . Moreover, we denote by the Hausdorff measure of noncompactness of , denote by the Hausdorff measure of noncompactness of and denote by the Hausdorff measure of noncompactness of .
Throughout this work, we suppose the following
The linear operator generates an equicontinuous -semigroup . Hence, there exists a positive number such that .
To discuss the problem (1), we also need the following lemma.
Lemma 5. If is bounded, then one has where .
Lemma 6 (see ). If is bounded, then for all , Furthermore, if is equicontinuous on , then is continuous on and
Since -semigroup is said to be equicontinuous, the following lemma is easily checked.
Lemma 7. If the semigroup is equicontinuous and , then the set for a.e. is equicontinuous for .
Definition 8. A function is said to be a mild solution of the nonlocal problem (1), if it satisfies
In addition, let be a finite positive constant, and set and .
3. Is Compact
In this section, we state and prove the existence theorems for the nonlocal impulsive problem (1). First, we give the following hypotheses:
(1) is a Carathéodory function; that is, for all is measurable and for a.e. is continuous;(2) for finite positive constant , there exists a function such that for a.e. and ;(3) there exists a function such that for a.e. and every bounded subset ;
() is a continuous and compact mapping; furthermore, there exists a positive number such that , for any ;
() is a continuous and compact mapping for every ;
Remark 9. The mapping is said to be -Carathéodory if the assumption is satisfied.
Theorem 10. If the hypotheses , , ,, and are satisfied, then the nonlocal problem (1) has at least one mild solution on .
To prove the above theorem, we need the following lemma.
Lemma 11. If the condition holds, then for arbitrary bounded set , we have
This proof is quite similar to that of Lemma 3.1 in ; we omit it.
Proof of Theorem 10. We consider the operator defined by
It is easy to see that the fixed points of are the mild solutions of nonlocal impulsive semilinear differential equation (1). Subsequently, we shall prove that has a fixed point by using Lemma 4.
We shall first prove that is continuous on . In fact, let be an arbitrary sequence satisfying in . It follows from Definition 8 that According to the continuity of in its second argument, for each , we have the following: In addition, and are all continuous for each , and hence, the Lebesgue dominated convergence theorem implies Namely, is continuous on .
Subsequently, we claim that . Actually, by , we obtain for any . Thus, .
Now we demonstrate that is equicontinuous for any . Let ,. Since is compact, is relatively compact; that is, there is a finite family such that for any , there exists some such that On the other hand, as is equicontinuous at , we can choose such that for each ,, uniformly for , and . By , it can be obtained that there exists such that for each ,, uniformly for . Furthermore, by Lemma 7, we get that there exists such that for each ,, uniformly for . Thus, there exists such that for each ,, uniformly for . Therefore, is equicontinuous at .
Similarly, we can conclude that is also equicontinuous at . Thus, is equicontinuous on .
Set . It is obvious that is equicontinuous on and maps into itself.
Next, we shall prove that is a convex-power condensing operator. Take ; by the definition of convex-power condensing operator, we shall show that there exists a positive integral such that if is not relatively compact. In fact, by using the conditions and , we get from Lemma 11 that Since , there exists a continuous function such that for any , Then where . Hence, Thus, and hence, by the method of mathematical induction, for any positive integer and , we obtain Therefore, for any positive integer , we have Since , it follows from the Stirling Formula (see ) that and hence, there exists sufficiently large positive integer such that which shows that is a convex-power condensing operator. From Lemma 4, we get that has at least one fixed point in ; that is, (1) has at least one mild solution . This completes the proof.
Remark 12. The technique of constructing convex-power condensing operator plays a key role in the proof of Theorem 10, which enables us to get rid of the strict inequality restriction on the coefficient function of . However, in many previous articles, such as [9, 12, 13, 19, 20], the authors had to impose a strong inequality condition on the integrable function , as they used Darbo-Sadovskii’s fixed point theorem only. Thus, our result extends and complements those obtained in [9, 12, 13, 19, 20] and has more broad applications.
Remark 13. If we use the following assumption instead of :
there exists a constant such that for a.e. and every bounded subset ,
we may use the same method to obtain for any . Thus, there exists a large enough positive integral such that namely, Therefore, we can get the following consequence.
Theorem 14. If the hypotheses , , ,, and are satisfied, then the nonlocal problem (1) has at least one mild solution on .
4. Is Lipschitz Continuous
In this section, by applying the proof of Theorem 10 and Darbo-Sadovskii’s fixed point theorem, we give the existence of mild solutions of the problem (1) when the nonlocal condition is Lipscitz continuous in .
We give the following hypotheses:
there exists a constant such that
is Lipschitz continuous with Lipschitz constant , for .
Theorem 15. If the hypotheses , , , , and are satisfied, then the nonlocal problem (1) has at least one mild solution on provided that .
Proof of Theorem 15. Given , let's first consider the following Cauchy initial problem: From the proof of Theorem 10, we can easily see that there exists at least one mild solution to (38). Define by that is the mild solution to (38). Then Now, we will show that is -condensing on . According to Lemma 11, for any bounded subset , we deduce which implies that In addition, since , it follows that the mapping is a -condensing operator on . In view of Lemma 2, the mapping has at least one fixed point in , which produces a mild solution for the nonlocal impulsive problem (1).
Remark 16. Similarly, one can show that the conclusion of Theorem 15 remains valid provided that hypothesis is replaced by condition .
Remark 17. In Theorem 15, we do not assume the compactness of nonlocal item . Under the Lipschitz assumption, we make full use of the conclusion of Theorem 10, the properties of noncompact measure and the technique of fixed point to deal with the solution operator .
Remark 18. Recently, the existence results for fractional differential equations have been widely studied in many papers. For more details on this theory one can refer to [29, 30] and references therein. It should be pointed out that the techniques and ideas in this paper can also be used to study fractional equations. In the future, we will also try to investigate to nonlocal controllability of impulsive differential equations by applying the similar techniques, methods, and compactness conditions. Further discussions on this topic will be in our consequent papers.
Example 1. Consider the following semilinear parabolic system: where is a bounded domain in with smooth boundary , is strongly elliptic, , and .
Let and define the operator by Then the operator is an infinitesimal generator of an equicontinuous -semigroup on (see ).
Suppose that the function satisfies the following conditions:(i)the Carathodory condition, that is, , is a continuous function about for a.e. is measurable about for each fixed ;(ii) for all with , where satisfies , uniformly in ;(iii) for all , where and .We assume the following.(1) is defined by Moreover, for given , there exist two integrable functions such that and for a.e. and every bounded subset ;(2) is defined by From Theorem 4.2 in , we get directly that is well defined and is a completely continuous operator by the above conditions about the function .(3) is a continuous and compact function for each defined by
Let us observe that the problem (42) may be reformulated as the abstract problem (1) under the above conditions. By using Theorem 10, the problem (42) has at least one mild solution ; provided that the hypothesis holds.
Example 2. Consider the following partial differential system: where is a bounded domain in with smooth boundary , and and both are given real numbers for .
Let , and define the operator by
As is known to all, the operator is an infinitesimal generator of the semigroup defined by for each . Here, is equicontinuous but is not compact.
We now suppose the following.(1) is defined by Moreover, for given , there exist two integrable functions such that and for a.e. , and every bounded subset ;(2) is defined by Then is Lipschitz continuous with constant ; that is, the assumption is satisfied.(3) is a continuous function for each , defined by Here we take ,,,,. Then is Lipschitz continuous with constant , ; that is, the assumption is satisfied.
Let us observe that (47) may be rewritten as the abstract problem (1) under the above conditions. If the following inequality holds, then according to Theorem 15, the impulsive problem (47) has at least one mild solution in .
The authors would like to thank the referees for their careful reading and their valuable comments and suggestions to improve their results. This research is supported by the Natural Science Foundation of China (11201410 and 11271316), the Natural Science Foundation of Jiangsu Province (BK2012260), and the Natural Science Foundation of Jiangsu Education Committee (10KJB110012).
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