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The periodic boundary value problem is discussed for a class of fractional evolution equations. The existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive mild solutions are obtained by using the monotone iterative technique. As an application that illustrates the abstract results, an example is given.

In this paper, we investigate the existence and uniqueness of positive mild solutions of the periodic boundary value problem (PBVP) for the fractional evolution equation in an ordered Banach space

The origin of fractional calculus goes back to Newton and Leibnitz in the seventieth century. We observe that fractional order can be complex in viewpoint of pure mathematics and there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proved to be valuable in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, electromagnetism, biology, and hydrogeology. For example space-fractional diffusion equations have been used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium [

Fractional evolution equations, which is field have abundant contents. Many differential equations can turn to semilinear fractional evolution equations in Banach spaces. For example, fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order

In this paper, without the assumptions of lower and upper solutions, by using the monotone iterative technique, we obtain the existence and uniqueness of positive mild solutions for PBVP (

We now turn to a summary of this work. Section

Let us recall the following known definitions. For more details see [

The fractional integral of order

The Riemann-Liouville derivative of order

The Caputo fractional derivative of order

(i) If

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

(iii) If

Throughout this paper, let

If

(i) See [

(ii) see [

(iii) see [

(iv) see [

(v) see [

(vi) see [

See [

For any fixed

If

We also introduce some basic theories of the operator semigroups. For an analytic semigroup

A

An analytic semigroup

For the applications of positive operators semigroup, we can see [

A bounded linear operator

By Remark

Let

For any

For sufficient conditions of exponentially stable operator semigroups, one can see [

If

For the applications of Lemma

Let

By (

Let

For any

There exists

Let

Set

Set

In the following, we prove that the uniqueness. If

Let

There exist

In Corollary

Consider the following periodic boundary value problem for fractional parabolic partial differential equations in

Let

Assume that

It is easy to see that

This research was supported by NNSFs of China (10871160, 11061031) and project of NWNU-KJCXGC-3-47.