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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 295837, 12 pages
Existence and Controllability Results for Fractional Impulsive Integrodifferential Systems in Banach Spaces
Department of Automation, China University of Petroleum, Beijing 102249, China
Received 4 March 2013; Revised 28 April 2013; Accepted 30 April 2013
Academic Editor: Rodrigo Lopez Pouso
Copyright © 2013 Haiyong Qin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We firstly study the existence of PC-mild solutions for impulsive fractional semilinear integrodifferential equations and then present controllability results for fractional impulsive integrodifferential systems in Banach spaces. The method we adopt is based on fixed point theorem, semigroup theory, and generalized Bellman inequality. The results obtained in this paper improve and extend some known results. At last, an example is presented to demonstrate the applications of our main results.
Fractional calculus is an area having a long history whose infancy dates back to three hundred years. However, at the beginning of fractional calculus, it develops slowly due to the disadvantage of technology. In recent decades, as the ancient mathematicians expected, fractional differential equations have been found to be a powerful tool in many fields, such as viscoelasticity, electrochemistry, control, porous media, and electromagnetic. For basic facts about fractional derivative and fractional calculus, one can refer to the books [1–4]. Since the fractional theory has played a very significant role in engineering, science, economy, and many other fields, during the past decades, fractional differential equations have attracted many authors, and there has been a great deal of interest in the solutions of fractional differential equations in analytical and numerical sense (see, e.g., [5–10] and references therein).
On the other hand, the impulsive differential systems are used to describe processes which are subjected to abrupt changes at certain moments [11–13]. The study of dynamical systems with impulsive effects has been an object of intensive investigations. It is well known that controllability is a key topic for control theory. Controllability means that it is possible to steer any initial state of the system to any final state in some finite time using an admissible control. We refer the readers to the survey  and the reference therein for controllability of nonlinear systems in Banach spaces. The sufficient controllability conditions for fractional impulsive integrodifferential systems in Banach spaces have already been obtained in [15–18].
Balachandran and Park  studied the controllability of fractional integrodifferential systems in Banach spaces without impulse where , the state takes values in the Banach space , , are continuous functions, and here . The control function , a Banach space of admissible control functions with as a Banach space, and is a bounded linear operator.
In , Mophou considered the existence and uniqueness of a mild solution for impulsive fractional semilinear differential equation where is the Caputo fractional derivative, and . The operator is a generator of -semigroup on a Banach space , and are impulsive functions.
To consider fractional systems in the infinite dimensional space, the first important step is to define a new concept of the mild solution. Unfortunately, By Hernández et al. , we know that the concept of mild solutions used in [15–17, 19], inspired by Jaradat et al. , was not suitable for fractional evolution systems at all. Therefore, it is necessary to restudy this interesting and hot topic again.
Recently, in Wang and Zhou , a suitable concept of mild solutions was introduced, using Krasnoselskii’s fixed point theorem and Sadovskii’s fixed point theorem, investigating complete controllability of fractional evolution systems in the infinite dimensional spaces where is the Caputo fractional derivative of the order with the lower limit zero, the state takes values in Banach space , and the control function is given in , with as a Banach space. is the infinitesimal generator of a strongly continuous semigroup in , is a bounded linear operator from to , and is given -value functions. Some sufficient conditions for complete controllability of the previous system were obtained.
Inspired by the work of the previous papers and many known results in [22–24], we study the existence of mild solutions for impulsive fractional semilinear integrodifferential equation where is the Caputo fractional derivative, , the state takes values in Banach space . is the infinitesimal generator of a strongly continuous semigroup of a uniformly bounded operator on , and is a bounded linear operator. is given -value functions, is defined as where are continuous, here , are impulsive functions, , , and and represent the right and left limits of at , respectively.
We also define a control and present controllability results for fractional integrodifferential systems in Banach spaces where is a bounded linear operator from to , and the control function is given in , with as a Banach space. The method we adopt is based on the ideas in [17–19, 22–24]. Compared with the previous results, this paper has three advantages. Firstly, we add operator in the nonlinear term and introduce a suitable concept of mild solutions of (4) and (6). Secondly, we not only study the existence of PC-mild solutions for impulsive fractional semilinear integrodifferential equation (4) but also present controllability results for fractional impulsive integrodifferential systems (6), and the results in [17, 19] could be seen as the special cases. Thirdly, our method avoids the compactness conditions on the semigroup , and some other hypotheses are more general compared with the previous research (see the conditions and ).
The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that are to be used later to prove our main results. In Section 3, the existence of PC-mild solutions for (4) is discussed. In Section 4, by introducing a class of controls, we present the controllability results for fractional impulsive integrodifferential systems (6). In Section 5, an example is given to illustrate the theory.
2. Preliminaries and Lemmas
Let us consider the set of functions , and there exist and , with . Endowed with the norm , it is easy to know that is a Banach space. Throughout this paper, let be the infinitesimal generator of a -semigroup of a uniformly bounded operators on . Let be the Banach space of all linear and bounded operator on . For a -semigroup , we set . For each positive constant , set .
Definition 1. The fractional integral of order with the lower limit zero for a function is defined as provided that the right side is point-wise defined on , where is the gamma function.
Definition 2. The Riemann-Liouville derivative of the order with the lower limit zero for a function can be written as
Definition 3. The Caputo derivative of the order for a function can be written as
Remark 4. (1) If , then
(2) The Caputo derivative of a constant is equal to zero.
(3) If is an abstract function with values in , then integrals which appear in Definitions 1, 2, and 3 are taken in Bochner’s sense.
Definition 5 (see ). A mild solution of the following nonhomogeneous impulsive linear fractional equation of the form is given by where and are called characteristic solution operators and given by and for , where is a probability density function defined on ; that is,
Definition 6. By a PC-mild solution of (4), we mean that a function , which satisfies the following integral equation:
Definition 7. By a PC-mild solution of the system (6), we mean that a function , which satisfies the following integral equation:
Definition 9 (see ). Let be a Banach space, and a one parameter family , , of bounded linear operators from to is a semigroup of bounded linear operators on if(1) (here, is the identity operator on );(2) for every (the semigroup property).A semigroup of bounded linear operator, , is uniformly continuous if .
Lemma 10 (see ). Linear operator is the infinitesimal generator of a uniformly continuous semigroup if and only if is a bounded linear operator.
Lemma 11 (see ). Let be a continuous and compact mapping of a Banach space into itself, such that is bounded. Then, has a fixed point.
Lemma 12. The operators and have the following properties. (i) For any fixed , and are linear and bounded operators; that is, for any , (ii) and are strongly continuous.(iii) and are uniformly continuous; that is, for each fixed , and , there exists such that
Proof. For the proof of (i) and (ii), the reader can refer to [23, Lemma 2.9] and [24, Lemmas 3.2–3.5]. For each fixed , and , one can obtain Because is a bounded linear operator, from Lemma 10 and Definition 9, we know that is the infinitesimal generator of a uniformly continuous semigroup. Thus, by the properties of uniformly continuous semigroup , we get that is, and are uniformly continuous.
We list here the hypotheses to be used later.() is continuous and there exist functions such that () is continuous and there exist function such that () There exist such that () The function is defined by where , , and , .() The constants and are defined by and , .
3. Existence of Mild Solutions
Theorem 13. If the hypotheses are satisfied, then the fractional impulsive integrodifferential equation (4) has a unique mild solution .
Proof. Define an operator on by
We will show that is well defined on . For , applying (28), we obtain
From the well-known inequality for and and Lemma 12, it is obvious that as . Thus, we deduce that .
For , we have It is easy to get that, as , the right-hand side of the previous inequality tends to zero. Thus, we can deduce that . By repeating the same procedure, we can also obtain that . That is, .
Take ; then, From and , we obtain So we deduce that In general, for each , , using the assumptions, when , obviously Noting that , with assumption and in the view of the contraction mapping principle, we know that has a unique fixed point ; that is, is a PC-mild solution of (4).
In order to obtain results by the Schaefer fixed point theorem, let us list the following hypotheses.() is continuous and there exist functions such that () is continuous and there exist functions such that () There exist such that () For all bounded subsets , the setis relatively compact in for arbitrary and , where () For all bounded subsets , the set is relatively compact in for arbitrary and .
Theorem 14. If the hypotheses are satisfied, the fractional impulsive integrodifferential equation (4) has at least one mild solution .
Proof. From Theorem 13, we know that operator is defined as follows:
We will prove the results in five steps.
Step 1 (continuity of on ). Let such that , and then and ; for every , we have Since the functions and are continuous, By conditions and , we know that Hence, By the Lebesgue dominated convergence theorem, we get It is easy to obtain that Thus, is continuous on .
Step 2 ( maps bounded sets into bounded sets in ). From (43), we get and we know that From (50) and (51), we obtain where . Thus, for any , Hence, we deduce that ; that is, maps bounded sets to bounded sets in .
Step 3. ( is equicontinuous with on ). For any , , we obtain after some elementary computation, we have Using the fact that and are uniformly continuous, and the well-known inequality for and , we can conclude that . Thus is equicontinuous with on .
Step 4 ( maps into a precompact set in ). We define and for . Set where From Lemma 12(ii)-(iii), , and the same method used in Theorem 3.2 of , we can verify that the set can be arbitrary approximated by the relatively compact set . Thus, is relatively compact in .
Step 5 (the set for some is bounded). Let , and thenSimilar to the results of (53), we know that Obviously there exists sufficiently small such that , and then we get Let It is clear that is nonnegative continuous function on , and generalized Bellman inequality implies that where is a constant. Obviously, the set is bounded on . Since is continuous and compact, thanks to Schaefer’s fixed point Theorem, has a fixed point (36) which is a PC-mild solution of (4).
4. Controllability Results
By introducing a class of controls, we present the controllability results for fractional impulsive integrodifferential systems (6).() The linear operator from into defined by induces an invertible operator defined on , and there exists a positive constant such that .
Theorem 15. If the hypotheses , , and are satisfied, then the fractional impulsive integrodifferential system (6) is controllable on .
Proof. Using the condition , for an arbitrary function , define the controlDefine the operator , whereBy Theorem 13, we know that is well defined, and we will prove that when using the previous control, operator has a fixed point. Clearly, this fixed point is a PC-mild solution of the control problem (6) and ; that is, the control we defined steers the system (6) from initial to in the time .
For any , by conditions , , and , we get Therefore, Since , then is contraction mapping. Any fixed point of is a PC-mild solution of (6) which satisfies . Thus, the system (6) is controllable on .
Theorem 16. If the hypotheses , , and are satisfied, the fractional impulsive integrodifferential system (6) is controllable on .
Proof. Using the condition , for an arbitrary function , define the controlWe will prove that when using the previous control, operator defined in (65) has a fixed point.
We discuss that in five steps.
Step 1 (continuity of on ). Let such that , and then and . For every , we have Since