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 [

On the other hand, the impulsive differential systems are used to describe processes which are subjected to abrupt changes at certain moments [

Balachandran and Park [

In [

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. [

Recently, in Wang and Zhou [

Inspired by the work of the previous papers and many known results in [

We also define a control

The rest of the paper is organized as follows. In Section

Let us consider the set of functions

The fractional integral of order

The Riemann-Liouville derivative of the order

The Caputo derivative of the order

(1) If

(2) The Caputo derivative of a constant is equal to zero.

(3) If

A mild solution of the following nonhomogeneous impulsive linear fractional equation of the form

By a PC-mild solution of (

By a PC-mild solution of the system (

The system (

Let

Linear operator

Let

The operators

For any fixed

For the proof of (i) and (ii), the reader can refer to [

We list here the hypotheses to be used later.

There exist

The function

where

The constants

If the hypotheses

Define an operator

For

Take

In order to obtain results by the Schaefer fixed point theorem, let us list the following hypotheses.

There exist

For all bounded subsets

is relatively compact in

For all bounded subsets

is relatively compact in

If the hypotheses

From Theorem

By introducing a class of controls, we present the controllability results for fractional impulsive integrodifferential systems (

The linear operator

induces an invertible operator

If the hypotheses

Using the condition

For any

If the hypotheses

Using the condition

We discuss that in five steps.

Consider the following nonlinear partial integrodifferential equation of the form