In this paper, we investigate a class of integral boundary value problems of fractional differential equations with a
In this paper, we consider the nonlinear fractional differential equations with a
In recent years, boundary value problems of fractional differential equations have significantly been discussed by some researchers because fractional calculus theory and methods have been widely used in various fields of natural sciences and social sciences. In the field of physical mechanics, fractional calculus not only provides suitable mathematical tools for the study of soft matter but also provides new research ideas and plays an irreplaceable role in the modeling of soft matter [
On the other hand, it is well known that differential equation models with
Moreover, during the last decade, the integral boundary value problem of fractional differential equations is also a hot issue for scholars and some good results have been achieved [
In [
Motivated by the works mentioned above, we concentrate on the solutions for the nonlinear fractional differential equation (
The rest of this paper is organized as follows. In Section
In the section, we present some definitions and lemmas, which are required for building our theorems.
Let
Suppose
Using the boundary condition
Thus,
From the above analysis, the equation
By (i) of Lemma (
Another, because
Now, we express
We obtain
Therefore,
Reverse, if
The proof is completed.
(1) For any
Therefore,
This completes the proof.
In this section, we will show the existence results for boundary value problem (
Let
(
(
Let
Define an operator
For any
By Lemma (
Thus,
We express
For each
As
In this section, we will give an example to illustrate our main results.
Consider the following equation:
Then,
It is obvious that
By Theorem (
No data were used to support this study.
The authors declare that they have no competing interests.
Both authors made an equal contribution.
This research is funded by the National Natural Science Foundation of China (No:11661037) and Scientific Research Fund of Jishou University (No:Jdy19004).