The Existence and Ulam Stability Analysis of a Multi-Term Implicit Fractional Differential Equation with Boundary Conditions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Report on the manuscript

‘Existence and Ulam stability analysis of multi term implicit fractional differential equation with boundary conditions’

by Peiguang Wang, Bing Han and Junyan Bao

 

In the present paper, the authors consider a class of multi term implicit fractional differential equation with boundary conditions. By using the fixed point theorems, the authors obtain the criterion of the existence, the existence and uniqueness theorems. Meanwhile, the criteria of Ulam-Hyers stability and Ulam-Hyers-Rassias stability are given. An example is given to illustrate the practical application and effectiveness of the results.

 

The whole proof process is rigorous, and the results are correct. This manuscript provides a useful contribution to the area of fractional differential equation. So I would like to recommend the paper for publication in Fractal Fract.

However, the flowing problems need to be corrected.

 

1. The manuscript should be reviewed carefully, to correct all the typing errors and grammatical errors. For example: In line 3 on page 1, “...allows us to establish a solution exists for the given equation” should be “...allows us to establish the criterion on a solution exists for the given equation”; In line 24-25 on page 2, “...to establish the existence result of Equation (1)” should be “.... to establish the existence results of Equation (1)”.

2. In the proof of Theorem 1, in line 37 on page 4, please further explain “Then use the following relationship exists between fractional integral and derivative” .

3. There are some results on existence and Ulam stability analysis of fractional differential equation in the existing literature compared with them, what is the main difference of the paper?

Comments on the Quality of English Language

Report on the manuscript

‘Existence and Ulam stability analysis of multi term implicit fractional differential equation with boundary conditions’

by Peiguang Wang, Bing Han and Junyan Bao

 

In the present paper, the authors consider a class of multi term implicit fractional differential equation with boundary conditions. By using the fixed point theorems, the authors obtain the criterion of the existence, the existence and uniqueness theorems. Meanwhile, the criteria of Ulam-Hyers stability and Ulam-Hyers-Rassias stability are given. An example is given to illustrate the practical application and effectiveness of the results.

 

The whole proof process is rigorous, and the results are correct. This manuscript provides a useful contribution to the area of fractional differential equation. So I would like to recommend the paper for publication in Fractal Fract.

However, the flowing problems need to be corrected.

 

1. The manuscript should be reviewed carefully, to correct all the typing errors and grammatical errors. For example: In line 3 on page 1, “...allows us to establish a solution exists for the given equation” should be “...allows us to establish the criterion on a solution exists for the given equation”; In line 24-25 on page 2, “...to establish the existence result of Equation (1)” should be “.... to establish the existence results of Equation (1)”.

2. In the proof of Theorem 1, in line 37 on page 4, please further explain “Then use the following relationship exists between fractional integral and derivative” .

3. There are some results on existence and Ulam stability analysis of fractional differential equation in the existing literature compared with them, what is the main difference of the paper?

Author Response

Please see the attachment file.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Please see the detailed report attached.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

I have mentioned in the attached detailed report.

Author Response

Please see the attachment file.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have updated the manuscript with care and answered all my previous comments satisfactorily.
I recommend the paper for publication in any high-level journal, especially in Fractal and Fractional.