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Controllability of Impulsive Neutral Functional Differential Inclusions in Banach Spaces
We investigate the controllability of impulsive neutral functional differential inclusions in Banach spaces. Our main aim is to find an effective method to solve the controllability problem of impulsive neutral functional differential inclusions with multivalued jump sizes in Banach spaces. Based on a fixed point theorem with regard to condensing map, sufficient conditions for the controllability of the impulsive neutral functional differential inclusions in Banach spaces are derived. Moreover, a remark is given to explain less conservative criteria for special cases, and work is improved in the previous literature.
During the last decade, differential inclusions [1–3] were well known for applications to mechanics, engineering, and so on. Impulsive differential equations [4–9] were important in the study of physical fields. Ahmed  first introduced three different models of impulsive differential inclusions and studied the existence of them, respectively. From then on, there have been many focuses on various properties of impulsive differential inclusions, see [11–17] and references therein.
Controllability is one of the primary problems in control theory [11, 13, 14, 17–24]. Study on controllability has always been considered as a hot topic given its numerous applications to mechanics, electrical engineering, medicine, biology, and so forth. Because of their various application backgrounds, there were a number of researches on controllability of differential inclusions, see [11, 13, 14, 17]. Controllability of impulsive functional differential inclusions is an attractive subject, thanks to their outstanding performance in applications. But as far as we are concerned, there were very few results on controllability of the model with multivalued jump sizes . As for the third model initiated by Ahmed , we were impressed by the statement that the model of differential inclusions with multi-valued jump sizes may arise under many different situations, for example, in case of a control problem where one wishes to control the jump sizes in order to achieve certain objectives. In this paper, we aim to find an effective method to solve the controllability problem of impulsive neutral functional differential inclusions with multi-valued jump sizes in Banach spaces.
Liu  studied impulsive neutral functional differential inclusions in Banach spaces. However, to the best of our knowledge, there has not any result considering the controllability of the impulsive neutral functional differential inclusions with multi-valued jump sizes in Banach spaces. This work is both challenging and interesting, since our systems are more general than those studied ever before. Based on a fixed point theorem with regard to condensing map, we work out the sufficient conditions for the controllability of impulsive neutral functional differential inclusions in Banach spaces. In , Liu considered the controllability basing on Martelli’s fixed point theorem . He took advantage of the statement that a completely continuous map is a condensing map. However, condensing map may not be completely continuous. We notice this inequality and consider the controllability on the strength of a special property of Kuratowski measure of noncompactness in Banach spaces. Due to the property, we are allowed to prove that a map is condensing according to its definition. When jumps are single-valued maps in our system, the system degenerates into the system (1.1) in . At this time, less conservative criteria can be given for controllability of system (1.1)  after appropriate degeneration. Work in  is improved.
The content of this paper is organized as follows. In Section 2, some preliminaries are recalled; the impulsive neutral functional differential inclusions is proposed. In Section 3, the results on controllability of impulsive neutral functional differential inclusions in Banach spaces are derived, as well as strictly proof; a remark is given to show our criteria are less conservative. In Section 4, conclusions are given to explain our work in this paper.
Definition 1. Let be a Banach space, a multi-valued map is called convex valued, if is convex for all .
is called closed valued, if is closed for all .
is called bounded on bounded set, if is bounded in for any bounded subset .
is called upper semicontinuous on , if for every , the set is a nonempty and closed subset of , and for every open set of containing , there is an open neighborhood of , such that .
We make the following notations: ; for any , is a bounded and measurable function on , and , where is a continuous function. Define norm on , as . is a Banach space .
, where are subsets of , ,
is bounded in ,
is convex in ,
is closed in ,
is bounded, convex, and closed in .
In this paper, we consider the neutral functional differential inclusions in Banach space as follows: where . For represents the defined by which belongs to some abstract phase space , where is a positive constant; is the infinitesimal generator of a strongly continuous operator semigroup ; is a closed, bounded, and convex valued multivalued map; is a continuous linear operator, where is a Banach space with , here is the control function; are closed, bounded, and convex valued multi-valued maps, , and represent the left and right limits of at , respectively. The histories .
We introduce definitions the following.
Definition 2. A function is called a mild solution of system (1) if the following holds: on and for each , the function is integrable, and there exists , such that the integral equation is satisfied, where , for a.e. .
Definition 4 (see ). A map is called -condensing, if for any bounded subset of , is bounded and , .
Remark 5. The in the Definition 4 is called the Kuratowski measure of noncompactness, which is defined as : there exist finitely many sets of diameter at most which cover . Measures of noncompactness are useful in the study of infinite-dimensional Banach spaces, where any ball of diameter has .
Lemma 6 (see ). Let be a Banach space, and is a condensing map. If the set is bounded, then has a fixed point.
3. Main Results
In order to study system (1), we introduce hypotheses hereinafter: is bounded, that is to say there are constants , such that . The linear operator defined by has an inverse operator , which takes value in . And is bounded. There exist positive constants and satisfying and . For each , there is a positive constant , such that for all . There exist constants , , , and , satisfying , and . is measurable with respect to for every , upper semicontinuous with respect to for every , and for every fixed is nonempty, or equivalently, . There is an integrable function and a continuous and nondecreasing function , such that .
Lemma 7 (see ). Let be a compact real interval, and let, be a Banach space. Let be a multivalued map satisfying , and let be a linear continuous mapping from . Then the operator is a closed graph operator in .
We denote the Banach space , , with seminorm defined by , where is the restriction of to .
Lemma 8. If and , then for , holds.
Proof. On the one hand, we have
On the other hand, .
The proof is thus completed.
If we define function as where , and denote , then is a mild solution of system (1) if and only if and where , .
Now we denote , and define norm , thus is a Banach space. Then we can define another multi-valued function , so where is the same as that in (10). Thus has a fixed point in if and only if has a fixed point in , .
Let , and is a positive constant. It is true that is a closed subspace of , so is also a Banach space. Next we show that has a fixed point in .
Lemma 9. is bounded, convex, and closed on .
Proof. (I) is bounded on . Let , thanks to Lemma 8,
Then we have
(II) is convex on . Let and , there must be and , such that for . Then for any , we have Combining and is convex, and is convex.
(III) is closed on . Let . Here we should proof that if there are sequences satisfying and , then holds.
For every , there is a and a , such that
And for , we should prove that there must be some and some , such that Considering calculation of (20) subtracting (19), we get Meanwhile is a closed multi-valued map, there truly exists some for (20).
Accordingly, (21) can be transformed to We construct a linear and continuous operator like moreover, From Lemma 7, is a closed graph operator. Then (22) implies that , with .
From the foregoing, (20) holds, which means . Thus is closed.
The proof is thus completed.
Proof. First, is a condensing map on . Considering Remark 5, we just have to prove that for any bounded subset of , . It is obvious, because diam for any by combining inequalities (19) and (25).
Second, here we show that the set for some is bounded. Let , then there are and , such that So . Due to Lemma 6, has a fixed point in . Consequently, has a fixed point, which means system (1) is controllable.
The proof is thus completed.
Remark 11. In case in system (1) are single-valued maps, then the system (1) degenerates into the system (1.1) in . Accordingly, our degenerated assumptions for the controllability Theorem 3.1  are less conservative, which means the following: firstly, the hypothesis  is unnecessary; secondly, the very complex hypothesis  can be replaced by inequality (25); finally, the assumption  is replaced by , so is not necessary to be bounded; otherwise, our results can only be applied to finite Banach spaces .
In this paper, we have investigated the controllability of impulsive neutral functional differential inclusions in Banach spaces. Based on a fixed point theorem with regard to condensing map, sufficient conditions for the controllability of the impulsive neutral functional differential inclusions in Banach spaces have been derived. Moreover, a remark has been given to explain less conservative criteria for special cases. We have found an effective method to solve the controllability problem of impulsive neutral functional differential inclusions with multi-valued jump sizes in Banach spaces. Work has been improved in the previous literature.
The authors would like to thank the editor and the reviewers for their constructive comments and suggestions which improved the quality of the paper. This work is supported by the National Science Foundation of China under Grant 61174039.
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