^{1}

^{1}

^{2}

^{1}

^{1}

^{1}

^{2}

By using the coincidence degree theory, we consider the following 2

Fractional differential equations have been of great interest recently. This is because of the intensive development of the theory of fractional calculus itself as well as its applications. Apart from diverse areas of mathematics, fractional differential equations arise in a variety of different areas such as rheology, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science (see [

However, there are few papers which consider the boundary value problem at resonance for nonlinear ordinary differential equations of fractional order. In [

In [

In [

In this paper, we study the 2

Setting

In this paper, we will always suppose that the following conditions hold:

We say that boundary value problem (

The rest of this paper is organized as follows. Section

Now, we will briefly recall some notation and an abstract existence result.

Let

The lemma that we used is [

Let

For the convenience of the reader, we present here some necessary basic knowledge and definitions about fractional calculus theory. These definitions can be found in the recent literature [

The fractional integral of order

The fractional derivative of order

We say that the map

for each

for almost every

for each

Assume that

We use the classical Banach space

For

Given

If

Let

Let

Let

Define

Let L be defined by (

In the following lemma, we use the unified notation of both for fractional integrals and fractional derivatives assuming that

Let

Then

Let

The mapping

Consider the continuous linear mapping

Recall (C2) and note that

Note that

Now,

Let

Note that the projectors

In fact, if

By Lemma

For every given

Assume that the following conditions on the function

There exist functions

There exists a constant

There exists a constant

or

Suppose (H1)-(H2) hold, then the set

Take

Now

If (

Suppose (H3) holds, then the set

Let

Suppose (H3) holds, then the set

We define the isomorphism

If the first part of (H3) is satisfied, let

Suppose the second part of (H3) holds, then the set

If (C1)-(C2) and (H1)–(H3) hold, then the boundary value problem (

Set

Finally, we will prove that (iii) of Lemma

Let us consider the following boundary value problem:

This work is sponsored by NNSF of China (10771212) and the Fundamental Research Funds for the Central Universities (2010LKSX09).