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A class of boundary value problems of Caputo fractional

Some researchers have paid close attention to the research of

There are also many papers about boundary value problems of fractional

Motivated by the methods of [

In Section

In this piece, we show some basic definitions and some lemmas that will be used to demonstrate our main results in the latter section.

Setting

The

If

We define the

For

The higher order

The

If

The higher order

We note that

Now let us give definitions of fractional

Let

The fractional

The fractional

Let

Let

Next, Green’s function for integral boundary value problem (

Given

By Lemma

The proof is complete.

It is obvious that

Suppose

According to the expression of

For the case

For the case

If

If

Assuming that

Our existence criteria will be based on the following Leggett-Williams fixed-point theorem.

Let

the set

Then

In this section, the above lemmas will be applied to obtain the main results of this paper.

Let

From Lemma

We shall use Lemma

Assume that condition (C) holds and there exist constants

Then the boundary value problem (

Set

We choose

Finally, for

Let condition (C) hold. Assume that there exist constants

Then the boundary value problem (

From (C4), we get

Let condition (C) hold. Assume that there exist constants

Then the boundary vale problem (

From hypothesis (C5), there exist

Assume that there exist two positive constants

Then the boundary value problem (

In line with (C6), it is easy to see that there exists a positive constant

In light of the proof of Theorems

Assume that the function

Assume that conditions (C), (C2), and (C3) hold. The function

Assume that conditions (C), (C1), and (C2) hold. The function

Assume that conditions (C), (C2), and (C4) hold. The function

Assume that conditions (C) and (C2) hold. The function

In this section, we present three examples to illustrate our results. We take

Consider the Caputo fractional

We take

All the conditions of Theorem

We take

All the conditions of Theorem

We seek

All the conditions of Theorem

The main innovation of this paper was that existence criteria of three positive solutions for a Caputo fractional

The authors declare that they have no conflicts of interest.

This project is supported by the National Natural Science Foundation of China (Grant no. 11271235), the Development Foundation of Higher Education Department of Shanxi Province (20101109, 20111117, and 20111020), and Shanxi Datong University Institute (2016K9 and 2017K4).