We mainly study the fractional evolution equation in an ordered Banach
space

In this paper, we use the perturbation theory and the monotone iterative technique based on lower and upper solutions to investigate the existence and uniqueness of mild solutions for the fractional evolution equation in an ordered Banach space

In particular, when

The fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) goes back to Newton and Leibnitz in the seventieth century. It has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields such as physics, chemistry, aerodynamics, viscoelasticity, porous media, electrodynamics of complex medium, and electrochemistry, control, electromagnetic. For instance, fractional calculus concepts have been used in the modeling of transmission lines [

One of the branches of fractional calculus is the theory of fractional evolution equations, that is evolution equations where the integer derivative with respect to time is replaced by a derivative of any order. Also, in recent years, fractional evolution equations have attracted increasing attention; see [

The monotone iterative technique based on lower and upper solutions is an effective and a flexible mechanism that offers theoretical as well as constructive existence results in a closed set. It yields monotone sequences of lower and upper approximate solutions that converge to the minimal and maximal solutions between the lower and upper solutions. Since under suitable conditions each member of the sequences happens to be the unique solution of a certain nonlinear problem, the advantage and importance of the technique is remarkable. For differential equations of integer order, many papers used the monotone iterative technique based on lower and upper solutions; see [

However, to the best of the authors' knowledge, no results yet exist for the fractional evolution equations by using the monotone iterative technique based on lower and upper solutions. Our results can be considered as a contribution to this emerging field.

In comparison with fractional ordinary differential equations, we have great difficulty in using the monotone iterative technique for the fractional evolution equations. Firstly, how to introduce a suitable concept of a mild solution for fractional evolution equations based on the corresponding solution operator? A pioneering work has been reported by El-Borai [

Our paper copes with the above difficulties, and the new features of this paper mainly include the following aspects. We firstly introduce a new concept of a mild solution based on the well-known theory of Laplace transform, and the form is very easy. Secondly, we discuss the perturbation properties for the corresponding solution operators. Thirdly, by the monotone iterative technique based on lower and upper solutions, we obtain results on the existence and uniqueness of mild solutions for problem (

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

The Riemann-Liouville fractional integral operator of order

The Caputo fractional derivative of order

For

We observe from the above that the Caputo fractional differential operators do not possess neither semigroup nor commutative properties, which are inherent to the derivatives on integer order. For basic facts about fractional integrals and fractional derivatives one can refer to the books [

Let

If

Similarly, we give the definitions of lower and upper solutions of problem (

If

Consider the following problem:

A family

In this case,

The solution operator

An operator

Let

Let

Let

Assume

For

If

A function

It is easy to verify that a classical solution of (

If

By (

Now, we discuss the perturbation properties of the solution operators.

An operator

From Definition

Assume that

By Lemmas

Assume that

If

If there is a

Assume

If

(i) If

(ii) If

If

By Definition

The linear Cauchy problem

By Lemmas

Assume that

Now, we recall some properties of the measure of noncompactness that will be used later. Let

Let

Let

There exists a constant

There exists a constant

Since

Let

Let

Even if

Let

Since

Let

Since

Let

Then the Cauchy problem (

We can find that

Therefore, by Theorem

By Corollaries

Let

Then the Cauchy problem (

Next, we consider the existence and uniqueness results of the Cauchy problem (

Let

There exists a constant

There exists a constant

Then the Cauchy problem (

Let

Let

Let

There exists a constant

Then the Cauchy problem (

By Corollaries

Let

In order to illustrate our main results, we consider the Cauchy problem in

For

There exist

The partial derivative

If (

From (

The author is grateful to the anonymous referee for his/her valuable suggestions and comments. This research supported by the Project of NWNU-KJCXGC-3-47.