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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 [

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 [

Chang et al. [

Balachandran et al. [

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:

We study the nonlocal initial problem (

Throughout this paper, let us consider the set of functions

The fractional integral of order

The Riemann-Liouville derivative of order

The Caputo derivative of order

If

Let

By a PC-mild solution of system (

By a PC-mild solution of system (

System (

Linear operator

Let

The operators

For any fixed

Since the infinitesimal generator

For

In order to prove the existence and uniqueness of mild solutions of (

The function

where

The constant

where

If hypotheses

Define the operator

For

In order to obtain more existence results, we have the following assumptions:

Define

There exists a function

Define

For all bounded subsets

is relatively compact in

For all bounded subsets

is relatively compact in

Let hypotheses

We shall present the results in six steps.

Define

From hypotheses we imposed and the same method used in [

Choose

It is sufficient to proceed exactly as in step 1 to step 4 of the proof to deduce that

In this section, we impose the following conditions to prove the results.

Define

induces an invertible operator

If hypotheses

Using

Assume that hypotheses

The proof is similar to that of Theorem

Consider the following nonlinear partial integrodifferential equation of the form

For

In this paper, we studied the existence and uniqueness results for a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions in a Banach space. Introducing the concept of PC-mild solutions and using the piecewise continuous control functions and uniformly continuous semigroup, we obtained the controllability results for the corresponding fractional impulsive integrodifferential system. Assuming that the semigroup is compact and utilizing some additional conditions, Hernández and O’Regan [

The authors declare that they have no conflicts of interest.

This research is supported by Shandong Provincial Natural Science Foundation (Grants nos. ZR2016AB04 and ZR2016JL021), a Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J17KB121), Major International (Regional) Joint Research Project of National Natural Science Foundation of China (Grant no. 61320106011), National Natural Science Foundation of China (Grants nos. 61503171 and 61527809), China Postdoctoral Science Foundation (Grant no. 2015M582091), Foundation for Young Teachers of Qilu Normal University (Grants nos. 2016L0605, 2017JX2311, and 2017JX2312), Doctoral Scientific Research Foundation of Linyi University (Grant no. LYDX2015BS001), and Scientific Research Foundation for University Students of Qilu Normal University (Grant no. XS2017L05).