Controllability of Impulsive Fractional Functional Integro-Differential Equations in Banach Spaces

This paper is concerned with the controllability problem for a class of mixed type impulsive fractional integro-differential equations in Banach spaces. Sufficient conditions for the controllability result are established by using suitable fixed point theorem combined with the fractional calculus theory and solution operator under some weak conditions.The example is given in illustrate the theory. The results of this article are generalization and improved of the recent results on this issue.


Introduction
In the past few decades, the fractional calculus, that is, calculus of integrals and derivatives of any arbitrary real or complex order has gained considerable popularity and importance, based on the wide applications in engineering and sciences such as fluid flow, rheology, dynamical processes in self-similar and porous structures, diffusive transport akin to diffusion, and electrical networks. For more details about fractional calculus theory and fractional differential equations with applications see the monographs of Baleanu et al. [1,2], Kilbas et al. [3], Lakshmikantham et al. [4], Miller and Ross [5], Podlubny [6], and the papers of [7][8][9][10][11][12][13][14].
Differential equations with impulsive conditions constitute an important field of research due to their numerous applications in ecology, medicine biology, electrical engineering, and other areas of science and technology. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments, see for instance the monographs by Lakshmikantham et al. [15], Bainov and Simeonov [16], Samoilenko and Perestyuk [17] and the papers of [18][19][20][21][22][23][24].
On the other hand, the controllability for fractional dynamical system has become an interesting research area to this field and one of the fundamental concepts in modern mathematical control theory. Very recently, the authors shu et al. [25] studied the existence of solutions for impulsive fractional differential equations, assuming the operator to be a sectorial, and the results are obtained by using Banach contraction theorem and Leray-schauder's alternative fixed point theorem. Shu et al. [26] established the existence and uniqueness of solutions for class of fractional partial semilinear functional differential equations with finite delay, here assuming is the infinitesimal generator of an analytic semigroup and by using Banach fixed point theorem.
Tomar and Dabas [27] extended the results of [25] into a controllability of impulsive fractional semilinear evolution equations with nonlocal conditions with as theresolvent family and the results are obtained by using Banach contraction principle. Many researchers [28][29][30][31][32][33][34][35][36][37][38][39] investigated the existence and controllability problem combined with fractional derivative with (or without) impulsive conditions. From above the collection of the literature survey, up till now, there is no work reported on this topic, and inspired by the above mentioned works [25][26][27]40] we will establish the controllability of impulsive fractional mixed type functional integro-differential equations with finite delay of the form where is Caputo fractional derivative of order 0 < < 1, : ( ) ⊂ → is the bounded linear operator of an -resolvent family { ( ) : ≥ 0} defined on a Banach space , : → is a bounded linear operator, Here, (⋅) represents the history of the time − , upto the present time . For ∈ , then ‖ ‖ = sup{| ( )| : We consider the problems (1)-(3) to study the controllability results using the solution operator and fixed-point theorems. The rest of this paper is organized as follows. In Section 2, we present some necessary definitions and preliminary results that will be used to prove our main results. The proof of our main results is given in Section 3. Finally, an example is included in Section 4.

Preliminaries
In this section, we mention some definitions and properties required for establishing our results. Let be a complex Banach space with its norm denoted as ‖ ⋅ ‖ , and ( ) represents the Banach space of all bounded linear operators from into , and the corresponding norm is denoted by ‖ ⋅ ‖ ( ) . Let ( , ) denote the space of all continuous functions from into with supremum norm denoted by ‖ ⋅ ‖ ( , ) . In addition, ( , ) represents the closed ball in with the center at and the radius .
A two-parameter function of the Mittag-Leffler type is defined by the series expansion where Ha is a Hankel path, that is, a contour which starts and ends at −∞ and encircles the disc | | ≤ | | 1/ contour clockwise. For short, ( ) = ,1 ( ). It is an entire function which provides a simple generalization of the exponent function: 1 ( ) = and the cosine function: 2 (− 2 ) = cos( ) and plays an important role in the theory of fractional differential equations. The most interesting properties of the Mittag-Leffler functions are associated with their Laplace integral see [3,6,41] for more details.
Definition 1 (see [40]). Caputo derivative of order for a function : [0, ∞) → is defined as for − 1 < < , ∈ . If 0 < ≤ 1, then The Laplace transform of the Caputo derivative of order > 0 is given as Definition 2 (see [42]). Let be a closed and linear operator with domain ( ) defined on a Banach space and > 0. Let ( ) be the resolvent set of . We call the generator of an -resolvent family if there exists ≥ 0 and a strongly continuous function : In this case, ( ) is called the -resolvent family generated by .

Journal of Function Spaces and Applications 3
Definition 3 (see [43]). Let be a closed and linear operator with domain ( ) defined on a Banach space and > 0. Let ( ) be the resolvent set of . We call the generator of an -resolvent family if there exists ≥ 0 and a strongly continuous function In this case, ( ) is called the solution operator generated by .
The concept of the solution operator is closely related to the concept of a resolvent family [44,Chapter 1]. For more details on -resolvent family and solution operators, we refer to [44,45] and the references therein.
From Lemma 4 we can verify that definition.
is called the reachable set of system (1) at terminal time .

Example
Consider the following fractional partial functional integrodifferential equations of the form where > 0, 0 < < 1. The the above example resembles the control system (1) where ( ) = √2/ sin( ), ∈ is the orthogonal set of eigen vectors of . It is well known that is the infinitesimal generator of an analytic semigroup { ( ) ≥0 } in and that is given by for all ∈ , and every > 0. From these expression it follows that, { ( ) ≥0 } is a uniformly bounded compact semigroup, so that, ( , ) = ( − ) −1 is a compact operator for ∈ ( ), that is, ∈ A ( 0 , 0 ).
(v) 1 : → is any function satisfying assumption (H 3 ). Therefore, the the system (36) can be written to the abstract form (1)- (3), and all the conditions of Theorem 7 are satisfied. We can conclude that the system (36) is controllable on .

Conclusions
In this article, abstract results concerning the controllability of impulsive fractional functional integro-differential equations involving Caputo fractional derivative in Banach spaces are obtained. By using fractional calculus theory and some standard fixed point theorem, we derived the controllability results. An example is provided to show the effectiveness of the proposed results.