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Existence of Mild Solutions and Controllability of Fractional Impulsive Integrodifferential Systems with Nonlocal Conditions
This paper is concerned with the existence results of nonlocal problems for a class of fractional impulsive integrodifferential equations in Banach spaces. We define a piecewise continuous control function to obtain the results on controllability of the corresponding fractional impulsive integrodifferential control systems. The results are obtained by means of fixed point methods. An example to illustrate the applications of our main results is given.
In recent decades, existence of mild solutions of nonlocal Cauchy problems has been investigated extensively by many researchers (see [1–15] and the references cited therein). The study of abstract nonlocal semilinear initial value problems was initiated by Byszewski and Lakshmikantham  and Byszewski . Byszewski  considered the existence and uniqueness of mild, strong, and classical solutions of nonlocal Cauchy problems. Lin and Liu  studied the existence and uniqueness of mild and classical solutions of semilinear integrodifferential equations with nonlocal Cauchy problems. Using Krasnoselskii’s fixed point theorem, Schauder’s fixed point theorem, and Banach contraction principle, Zhou and Jiao  obtained several criteria on the existence and uniqueness of mild solutions of nonlocal Cauchy problems for fractional evolution equations without impulse.
Such analysis on nonlocal Cauchy problems is important from an applied viewpoint, since the nonlocal condition has a better effect in applications than a classical initial one. For instance, the diffusion phenomenon of a small amount of gas in a transparent tube can be given a better description than using the usual local Cauchy problem. On the other hand, controllability of nonlocal problems in Banach spaces has become an active area of investigation; we refer the reader to, for example, the papers [16–29]. The most common method is to transform the controllability problem into a fixed-point problem of solutions for an appropriate operator in a function space, that is, the existence problem of differential and integrodifferential equations. Unfortunately, by , we know that the concept of mild solutions used in [14, 15, 17] was not suitable for fractional evolution systems.
Chang et al.  investigated the controllability of a class of first-order semilinear differential systems with nonlocal initial conditions in a Banach space: where generates a strongly continuous, not necessarily compact, semigroup in the Banach space . Sufficient conditions for the controllability of the first-order semilinear differential system with nonlocal initial conditions were established. The approach used is Sadovskii’s fixed point theorem.
Balachandran et al.  discussed the controllability of a class of fractional integrodifferential systems with nonlocal conditions in a Banach space:
Motivated by the work of the above papers and wide applications of nonlocal Cauchy problems in various fields of natural sciences and engineering, in this paper, we study the existence of nonlocal problems for a class of fractional impulsive integrodifferential systems in a Banach space of the following type: where and is the Caputo fractional derivative (); the state takes values in the Banach space . is the infinitesimal generator of a strongly continuous semigroup of uniformly bounded operators in , and is a bounded linear operator. is a given -value function; is continuous; here , , , , , and represent the right and left limits of at , respectively. Using the similar method and a piecewise continuous control function, we consider the controllability of a class of fractional impulsive integrodifferential systems with nonlocal initial conditions: where is a bounded linear operator from to and the control function is given in , with as a Banach space.
We study the nonlocal initial problem (3) that describes a more general form than the previous ones reported in [18, 19]. We introduce a suitable concept of PC-mild solutions for nonlocal initial problem (3). We not only study the existence and uniqueness of a mild solution for impulsive fractional semilinear integrodifferential equation (3) but also define a piecewise continuous control function and present the results on the controllability of the corresponding fractional impulsive integrodifferential system (4) which include some known results obtained in [14, 17]. Assumptions in our results are less restrictive.
2. Preliminaries and Lemmas
Throughout this paper, let us consider the set of functions and there exist and , , with . Endowed with the norm , it is easy to verify that is a Banach space. Let be the Banach space of all linear and bounded operators on . For a -semigroup , we set . For each positive constant , we set . Obviously, is a bounded closed and convex subset.
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 order with the lower limit zero for a function can be written as
Definition 3. The Caputo derivative of order for a function can be written as
Definition 5 (see ). Let be a Banach space; a one-parameter family , , of bounded linear operators from to is a semigroup of bounded linear operators on if (1); is the identity operator on ;(2) for every (the semigroup property). A semigroup of bounded linear operators, , is uniformly continuous if .
Definition 6 (see ). By a PC-mild solution of system (3), we mean a function that satisfies the following integral equation: where and are called characteristic solution operators and are given by and, for ,where is a probability density function defined on ; that is,
Lemma 9 (see ). Linear operator is the infinitesimal generator of a uniformly continuous semigroup if and only if is a bounded linear operator.
Lemma 10 (see  Krasnoselskii's fixed point theorem). Let be a Banach space, let be a bounded closed and convex subset of , and let be maps of into such that for every pair . If is a contraction and is completely continuous, then the equation has a solution in .
Lemma 11 (see [22, 23]). The operators and defined by (9) 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.
Lemma 13 (see ). For and , .
3. Existence and Uniqueness of PC-Mild Solutions
In order to prove the existence and uniqueness of mild solutions of (3), we have the following assumptions: is continuous and there exist two functions such that is continuous and there exists a function such that are continuous and there exist such that is continuous and there exists a function such that The function is defined by where , , and , .()The constant and function are defined by where and , .
Theorem 14. If hypotheses – are satisfied, then (3) has a unique PC-mild solution.
Proof. Define the operator on by For , by virtue of (20), we conclude thatIt follows from Lemma 11, part (iii) and Lemma 13 thatThus, we deduce that . For , we have From (23), we know that . Using the same method, we obtain ,, , and therefore . For each , , When , we getIt follows now from , , and the contraction mapping principle that has a unique fixed point ; that is, is a unique PC-mild solution of (3). The proof is complete.
In order to obtain more existence results, we have the following assumptions: is continuous and there exist three functions such that is continuous and there exist two functions such that are continuous and there exist such that Define There exists a function such that Define For all bounded subsets , the set is relatively compact in for arbitrary and , where and are defined by For all bounded subsets , the set is relatively compact in for arbitrary and .
Theorem 15. Let hypotheses and – be satisfied. If the inequalities hold, where and is as in , then (3) has at least one PC-mild solution.
Proof. We shall present the results in six steps.
Step 1 (Continuity of defined by (20) on ). Let and . Then and . For , we have Since the functions , , and are continuous, we conclude that Applications of and yield which implies that By Lebesgue’s dominated convergence theorem, we get and so Step 2 ( maps bounded sets into bounded sets in . From (20), we get where By Lemma 11 and (42), we obtain Thus, for any , we have Hence, we deduce that , that is, maps bounded sets into bounded sets in .
Step 3 ( is equicontinuous with on . For any , , , we obtain Based on a straightforward computation, we have It follows from Lemma 11, part (iii) and Lemma 13 that . Thus, is equicontinuous with on .
Step 4 ( map into a precompact set in ). We define the operator where Define and for . Set where From hypotheses we imposed and the same method used in [16, Theorem3.2], it is not difficult to verify that the set can be arbitrary approximated by the relatively compact set . Thus, are relatively compact in .
Step 5 ( for ). Note that Choose and define operators and on by It is sufficient to proceed exactly as in step 1 to step 4 of the proof to deduce that are continuous and compact. Thus, to complete this proof, it suffices to show that is a contraction mapping and that for . Indeed, for any , by virtue of (43) and (51), we have