We study the existence of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations
with nonlocal conditions in a separable Banach space

In this paper, we denote that

The domain

The operator

There exist constants

Under condition (A2), each operator

Moreover,

This paper is concerned with existence result for nonautonomous fractional integrodifferential equations with nonlocal conditions in a separable Banach space

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The field of the application of fractional calculus is very broad. We can see it in the study of the memorial materials, earthquake analysis, robots, electric fractal network, fractional sine oscillator, electrolysis chemical, fractional capacitance theory, electrode electrolyte interface description, fractal theory, especially in the dynamic process description of porous structure, fractional controller design, vibration control of viscoelastic system and pliable structure objects, fractional biological neurons, and probability theory. For details, see the monographs of Kilbas et al. [

Moreover, since the work of Byszewski [

In this paper, using a pair of evolution families

Throughout this paper, we set

We set

Next, we recall the definition of the Riemann-Liouville integral.

The fractional (arbitrary) order integral of the function

We have (1)

(2) Obviously, for

The Riemann-Liouville derivative of order

Based on the work in [

Let

By using the family

The operator-valued functions

Moreover, we have

A mild solution of (

A function

We will need the following facts from the theory of measures of noncompactness and condensing maps (see, e.g., [

Let

As an example of the MNC, we may consider the Hausdorff MNC:

We know that

for any

for any

for every relatively compact set

for each

In Section

Let

integrable, if it admits a Bochner integrable selection

integrably bounded, if there exists a function

For an integrable, integrably bounded multifunction

Let

A continuous map

The application of the topological degree theory for condensing maps (see, e.g., [

Let

We need the hypotheses as follows.

Function

for almost all

There exists a function

where

The function

Define the operator

The operator

Let

Therefore, the fact

The operator

For any

So, the set

The operator

Noting that, for any

For every bounded subset

Let

For any

We consider the multifunction

Therefore, combining with (

For any

Now, from (

Further, from Proposition

Assume that (H1), (H2), and (H3) are satisfied; then problem (

Let us introduce in the space

Next, we show that there exists some

Combining with (H1)–(H3), Remark

Dividing both sides of (

In this section, set

For

Then (

Moreover,

For

This paper deals with the existence of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with nonlocal conditions in an abstract space. Sufficient conditions for the existence of mild solution are derived with the help of the fixed point theorem for condensing maps. An example is provided to illustrate the obtained result.

The author is grateful to the referees for their valuable suggestions. This work was partly supported by the NSF of China (11201413), the NSF of Yunnan Province (2009ZC054M, 2013FB034), the Educational Commission of Yunnan Province (2012Z010), and the Foundation of Key Program of Yunnan Normal University.