We study the periodic boundary value problem for semilinear fractional differential equations in an ordered Banach space. The method of upper and lower solutions is then extended. The results on the existence of minimal and maximal mild solutions are obtained by using the characteristics of positive operators semigroup and the monotone iterative scheme. The results are illustrated by means of a fractional parabolic partial differential equations.

In this paper, we consider the periodic boundary value problem (PBVP) for semilinear fractional differential equation in an ordered Banach space

Fractional calculus is an old mathematical concept dating back to the 17th century and involves integration and differentiation of arbitrary order. In a later dated 30th of September 1695, L’Hospital wrote to Leibniz asking him about the differentiation of order 1/2. Leibniz’ response was “an apparent paradox from which one day useful consequences will be drawn.” In the following centuries, fractional calculus developed significantly within pure mathematics. However, the applications of fractional calculus just emerged in the last few decades. The advantage of fractional calculus becomes apparent in science and engineering. In recent years, fractional calculus attracted engineers’ attention, because it can describe the behavior of real dynamical systems in compact expressions, taking into account nonlocal characteristics like infinite memory [

In recent years, there have been some works on the existence of solutions (or mild solutions) for semilinear fractional differential equations, see [

However, to the authors’ knowledge, no studies considered the periodic boundary value problems for the abstract semilinear fractional differential equations involving the operator semigroup generator

The method of upper and lower solutions has been effectively used for proving the existence results for a wide variety of nonlinear problems. When coupled with monotone iterative technique, one obtains the solutions of the nonlinear problems besides enabling the study of the qualitative properties of the solutions. The basic idea of this method is that using the upper and lower solutions as an initial iteration, one can construct monotone sequences, and these sequences converge monotonically to the maximal and minimal solutions. In some papers, some existence results for minimal and maximal solutions are obtained by establishing comparison principles and using the method of upper and lower solutions and the monotone iterative technique. The method requires establishing comparison theorems which play an important role in the proof of existence of minimal and maximal solutions. In abstract semilinear fractional differential equations, positive operators semigroup can play this role, see Li [

In Section

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

If

We need some basic definitions and properties of the fractional calculus theory which are used further in this paper.

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

For more fractional theories, one can refer to the books [

Throughout this paper, let

If

If

(i) [

(ii) [

(iii) [

If

In

For sufficient conditions of exponentially stable

A

A

Compact semigroups, differential semigroups, and analytic semigroups are equicontinuous semigroups, see [

A

From Definition

A bounded linear operator

The operators

For any fixed

For the proof of (i)–(iii), one can refer to [

For

By Remark

By

Assume that

There exists a constant

Then PBVP (

It is easy to see that

Let

In the following, we prove that

Furthermore, for

Then by Ascoli-Arzela’s theorem,

Assume that

There exists a constant

Then PBVP (

Let

Assume that the positive cone

By the proof of Theorem

For

By Ascoli-Arzela’s theorem,

Let

In an ordered and weakly sequentially complete Banach space, the normal cone

Let

In an ordered and reflective Banach space, the normal cone

By Theorem

Assume that _{1}) and one of the following conditions:

then PBVP (

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

Let

Let

there exists a constant

Then PBVP (

Set

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