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We use Sadovskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order

Dynamics of many evolutionary processes from various fields such as population dynamics, control theory, physics, biology, and medicine. undergo abrupt changes at certain moments of time like earthquake, harvesting, shock, and so forth. These perturbations can be well approximated as instantaneous change of states or impulses. These processes are modeled by

The origin of fractional calculus (derivatives

Fractional derivatives arise naturally in mathematical problems, for

One can see that both of the fractional derivatives are actually nonlocal operator because integral is a nonlocal operator. Moreover, calculating time fractional derivative of a function at some time requires all the past history, and hence fractional derivatives can be used for modeling systems with memory. Fractional differential equations can be formulated using both Caputo and Riemann-Liouville fractional derivatives. A

Our main objective is to discuss existence and uniqueness of solutions of the following

Now, we define some important spaces and norm which will encounter frequently:

A solution of fractional differential equation (

Kuratowskii noncompactness measure: let

Condensing map: Let

Compact map: a map

Let B be a convex, bounded, and closed subset of a Banach space

A map

(a)

(b)

and hence

The structure of the paper is as follows. In Section

Consider the initial value problem (

The initial value problem (

Under the assumptions

Let

And hence by using the Theorem

If

In this section, we discuss existence and uniqueness of solutions of the following

Define a new Banach space

with sup-norm

A solution of fractional differential equation (

Consider the initial value problem (

The initial value problem (

Since history part/initial condition

Under the assumptions (

In this case, we define the operator

The proof is similar to the proof of continuity of

Now by applying Banach’s fixed point theorem, we get that the operator

Our next result is based on Schaefer’s fixed point theorem. In this case, we replace assumption (

Under the assumptions ^{'}^{'}

We transform the problem into a fixed point problem. For this purpose, consider the operator

For

As a consequence of Steps 1–3 together with PC-type Arzela-Ascoli theorem (Fečken et al. [

We observe that for

As a consequence of Schaefer’s fixed point theorem, the problem (

If

Further, we consider the following more general Caputo fractional differential equation

A solution of fractional differential equation (

Consider the initial value problem (

The initial value problem (

Under the assumptions

The proof is similar to the proof of Theorem

Under the assumptions ^{'}^{'}

The proof is similar to the proof of Theorem

If

Consider the following class of fractional logistic equations in banach space

The initial value problem (

We can easily see that for the problem (

Also

Using Gronwall’s inequality (Diethelm, Lemma 6.19 [

We give some more examples which are inspired by [

Consider the following Caputo impulsive delay fractional differential equations

Set

Let

Again, for all

For

Consider the following Caputo impulsive delay fractional differential equation

Set

Again, for all

For

The authors are thankful to the anonymous reviewer for his/her valuable comments and suggestions. S. Abbas acknowledge “Erasmus Mundus Lot13, India 4EU” for providing him fellowship to visit University of Bologna, Italy. He also acknowledges Professor Stefan Siegmund, Center for Dynamics of TU Dresden, Germany, for his fruitful discussion on this topic and for hosting his short visit.