We study the existence and nonexistence of the positive solutions for the integral boundary value problem of the fractional differential equations with the disturbance parameter

In this paper, we are concerned with the existence and nonexistence of positive solutions for the boundary value problem of the fractional differential equations

In the recent decades, since fractional differential equations have been applied widely and successfully in the description of complex dynamics, they have been regarded as a valuable tool being used in the fields to handle viscoelastic, physics, chemistry, electrical engineering, biology aspects, and so forth; see [

At the same time, while using the methods of the differential equations to solve the actual problems, it is inevitable that there always exists disturbance which will have great influence on the existence of the solutions. In paper [

The purpose of this paper is to study the impact of the disturbance parameter

This paper is organized as follows. In Section

In this section, we give some basic definitions and lemmas which play an important role in our research.

Let

The Caputo derivative of order

If

Let

If

Suppose

One says

Throughout this paper, we assume the following conditions hold.

There exists a function

For

Suppose

According to Lemma

By the boundary condition

Hence, we can obtain

It follows the definition Riemann-Liouville fractional integral that

We multiply by the function

Hence,

It follows from (

We can obtain

Similarly, we can obtain the following lemma.

Suppose

Suppose

If

For convenience, we denote

Similar to the proof of Lemma

Substituting

We multiply by the function

On the other hand, if

Hence,

We can easily verify that

Therefore,

Suppose

Since (H1) holds, we can show that for

It is easy to see that

For

We denote

By (

Hence,

For the sake of the reader, we state the fixed point index theorem and Schauder’s fixed point theorem which will be used later.

Let

Let

Let

Let

The following comparison principle will play a very important role in our main results analysis.

Let

Denote

By Lemma

It follows that

Let

Let

Since (H2) holds, we have

Hence, by (

We can easily obtain the following lemma from the definition of

Suppose

Let

Suppose

By Lemma

In view of Lemma

We define

By Lemma

Suppose (H1), (H2), and (H3) hold and there exist a nonnegative lower solution

Let

By Lemma

We define

Then

Next we can prove that

Let

We denote

Since

By (H1), it is easy to see

Therefore, as

Thus, we have proved

By Arzela-Ascoli theorem, we know that

We can easily show that

Since

Then

Finally, we prove

We can prove that if each solution

Let

If

It follows that

If

By (H3), we can get

It follows that

Hence, we show

Similarly, we can get

Therefore, each solution

Suppose (H1), (H2), and (H3) hold:

Since

We can easily verify that

By Theorem

Therefore, if there exists a constant

For convenience, we give the following notations:

We can see that

Suppose that

Since

Since

By Lemma

In view of Lemma

We take

Suppose (H1) holds. If one of the following conditions is satisfied, then the boundary value problem (

there exist constants

Because

We take

By

Suppose (H1) holds

If

If

Let

Let

For any

Let

We choose

Suppose that (H1), (H2), and (H3) hold. If

We have the following four steps to prove the conclusions of Theorem

For each

Then, in view of Theorem

It follows that

Let

Since

Let

In view of Lemma

By Lebesgue dominated convergence theorem, we can get

Hence,

Let

For each

Let

Similarly, let

In fact, we can easily show that

Let

We choose

Define

Similar to the proof of Theorem

By Lemma

Let

Since

Because

Since

We define

It is obvious that

We can prove

Otherwise, if there exists

According to the homotopy invariance of the fixed point index and (

By the additivity property of the fixed point index, (

Therefore, the boundary value problem (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful to the referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (no. 11171220) and the Foundation of the Education Department of Shanghai (no. 10ZZ93).