Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump

Round 1

Reviewer 1 Report

This paper is very clear, well-written and with important results. I would only suggest to the authors the following:

- Please outline the difference between uncertain fractional differentail equations with jump with (general) stochastic processes. This would clarify the context for a wide audience.

- Since you deal with on jump processes , you should also mention CADLAG functions and their place in the global framework that you consider, again in order to be more clear for a wide audience.

These clarifications can be made in very few lines, in the introduction.

Beyond all that, I address to you my best congratulations for this work which is very interesting.

Author Response

            Responses to Reviews on Symmetry-781954

Title: Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump

Dear Reviewer:

Thank you for your comments concerning our manuscript entitled “Symmetry-781954”.  Those comments are all valuable and very helpful for revising and improving our paper, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with your approval. Let us now answer your comments one by one as follows:

Responses to Reviews #1:

• Q1: Please outline the difference between uncertain fractional differential equations with jump with (general) stochastic processes. This would clarify the context for a wide audience.

A1：Thanks for your positive evaluations and your approvals of our manuscript. As you advised, we add the first paragraph of the introduction part. We outline the difference between uncertain processes and stochastic processes. We add “Wiener process is a type of stationary-independent increment stochastic process with normal random increments designed by Wiener in 1923 [1]. Then stochastic differential equation (SDE, for short) was proposed by Itô in 1951 as a vital tool to model stochastic dynamic systems [2]. Following that, many areas such as noted European option pricing model [3] by SDEs and famous stochastic epidemic dynamic model hidden in the observed data [4] were developed. As all we know, The SDEs based on probability theory need a large of available sample data. However, when we lack of data or the size of sample data applied in practice are less in many situations, we need to invite some domain experts to evaluate the belief degree that each event happens.” on page 1 in the introduction part.

• Q2:Since you deal with on jump processes , you should also mention CADLAG functions and their place in the global framework that you consider, again in order to be more clear for a wide audience.

These clarifications can be made in very few lines, in the introduction.

Beyond all that, I address to you my best congratulations for this work which is very interesting.

A2：Thanks for your positive evaluations and your approvals of our manuscript. As you advised, we add CADLAG functions in the introduction part.

Finally, many thanks for your positive evaluations of our manuscript again.

Reviewer 2 Report

Please see the comment as attached.

Comments for author File: Comments.pdf

Author Response

                      Responses to Reviews on Symmetry-781954

Title: Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump

Dear Reviewer:

Please see the responses as attached. 

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The manuscript has been revised and the quality has improved.