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Using Krein-Rutman theorem, topological degree theory, and bifurcation techniques, this paper investigates the existence of positive solutions for a class of boundary value problems of fractional differential inclusions.

Fractional differential equations have been of great interest recently. Engineers and scientists have developed new models that involve fractional differential equations. These models have been applied successfully, for example, in mechanics (theory of viscoelasticity and viscoplasticity), (bio)chemistry (modelling of polymers and proteins), electrical engineering (transmission of ultrasound waves), medicine (modelling of human tissue under mechanical loads), and so forth. For details, see [

In [

In this paper, we consider the following boundary value problem of fractional differential inclusions of the form

As mentioned in [

Also there are some papers concerned with initial or boundary value problems of fractional differential inclusions (see, for instance, [

The paper is organized as follows. Section

For convenience, we present some necessary definitions and results from fractional calculus theory (see [

The fractional (arbitrary) order integral of the function

For a function

Let

Assume that

The relation

For more detailed results of fractional calculus, we refer the reader to [

Let

Let

The following lemmas are crucial in the proof of our main result.

Let

For more details on multivalued maps, see the books of Deimling [

Finally in this section, we list the following results on topological degree of completely operators.

Let

Then there exists a connected component

Let

there exists an interval

Let

Then

Suppose that the following two assumptions hold throughout the paper.

(H1) Let

(H2) There exist functions

The basic space used in this paper is

We first consider the following linear boundary problem of fractional differential equation:

Given

The function

For the sake of using bifurcation technique to investigate BVP (

A function

For

From Lemmas

The operator defined by (

Notice that the operator

Note that condition (H1) implies that

Define

Let

For

Let

Suppose, on the contrary, that there exist

On the other hand, from (H2) we know

To sum up,

For

Notice that

Therefore, by the homotopy invariance of topological degree, we have

For

We first prove that for

Suppose, on the contrary, that there exist

By Lemma

Letting

The conclusion of this lemma follows.

From Lemma

For fixed

By a process similar to the above, one can obtain the following conclusions.

Let

For

For

Finally, using Lemmas

The main results of this paper are the following two conclusions.

Suppose that (H1) and (H2) hold. In addition, suppose either

Then BVP (

We need only to prove that there is a component of

From Theorem

If

If

From Theorem

We show that

Suppose that (H1), (H2), and the following assumption holds.

There exist

In addition, suppose

Then BVP (

From Theorems

For this sake, from assumption (H3), there exists

Immediately,

Notice that

Consequently, BVP (

Let

Consider the following boundary value problem of fractional differential inclusions

Then BVP (

BVP (

By the definition of

Therefore, by Theorem

The research is supported by NNSF of China (11171192) and the Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2010SF025).