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We study the existence and multiplicity of solutions for the following fractional boundary value problem:

Consider the fractional boundary value problem (BVP for short) of the following form:

Differential equations with fractional order are generalization of ordinary differential equations to noninteger order. This generalization is not mere mathematical curiosities but rather has interesting applications in many areas of science and engineering such as in viscoelasticity, electrical circuits, and neuron modeling. The need for fractional order differential equations stems in part from the fact that many phenomena cannot be modeled by differential equations with integer derivatives. Such differential equations got the attention of many researchers and considerable work has been done in this regard, see the monographs of Kilbas et al. [

Recently, there are many papers dealing with the existence of solutions (or positive solutions) of nonlinear initial (or singular and nonsingular boundary) value problems of fractional differential equation by the use of techniques of nonlinear analysis (fixed-point theorems [

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In [

In this paper, in order to establish the existence and multiplicity of solutions for BVP (

For the superquadratic case, we make the following assumptions.

where

Assume that (A1)–(A3) hold and that

For the asymptotically quadratic case, we assume the following.

There exists

Our second and third main results read as follows.

Assume that

Assume that

there exists

For the subquadratic case, we give the following multiplicity result.

Assume that

Then BVP (

In this section, we recall some background materials in fractional differential equation and critical point theory. The properties of space

Let

Let

The left and right Caputo fractional derivatives are defined via the above Riemann-Liouville fractional derivatives. In particular, they are defined for the function belonging to the space of absolutely continuous functions, which we denote by

Let

The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, that is,

Define

Let

Let

According to (

Define

Making use of Property

In the following, we will treat BVP (

Let

If

A function

Let

By Lemma

If

Let

Let

there exist

there exists

Let

For

It follows from Lemma

Recall that a sequence

We will first establish the following lemma and then give the proof of Theorem

Assume (A), (A2), and (A3) hold, then the functional

Let

By (A2), there exist positive constants

It follows from

Combining (

On the other hand, by (A3), there exist

By (A), we have

Therefore, we obtain

It follows from (

If

If

Since

By Proposition

Then we obtain

By (A1), there exist

Let

Therefore, we have

It is obvious from the definition of

By (A1), there exist

It follows from (A) that

Therefore, we obtain

Choosing

For

Finally, noting that

The following lemmata are needed in the proof of Theorem

Assume (A5), then for any

The proof is similar to that of Lemma 2 in [

Assume (A), (A2’), (A4), and (A5), then the functional

Suppose that

We only need to show that

By (A2), it follows that there exist constants

By assumption (A), it follows that

Passing to the limit in the last inequality, we get

By virtue of Lemma

We assert that meas

Since

By (A4), we obtain thye following:

By (

By virtue of Lemmas

The proof of Theorem

The functional

By (

The functional

Let

Suppose that

However, from (

Noting that

For any

For every

For any

Since

It is easy to prove that the odd mapping

Since

Otherwise, for any positive integer

Set

Since

By the equivalence of the norms on the finite dimensional space, we have

By (

Thus, there exist

In fact, if not, we have

It implies that

Now let

By (

For any

Choosing

Now from the assertion of Lemma

In this section, we give some examples to illustrate our results.

In BVP (

These show that all conditions of Theorem

By Theorem

In BVP (

Let

If

In BVP (

By Theorem

This paper is partially supported by the NNSF (no. 11171351) of China.