Existence and Uniqueness Theorems for a Variable-Order Fractional Differential Equation with Delay

Round 1

Reviewer 1 Report

This work has been an interesting read and I believe the results are relevant to the research community.

 

The introduction is well written with adequate references to relevant literature.

 

Theorems, Definitions and Lemmas have been well presented in the work with proofs where needed.

 

Below are a few comments for the authors.

 

The article has little to no grammatical errors. A final review should correct any errors.

 

The sentence in the conclusion below is unclear. Is there a word after “to”?

“For both the theoretical and numerical components, this approach can also be used to____ FDEs with variable order and multi-point boundary conditions.”

 

 

Author Response

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Author Response File: Author Response.docx

Reviewer 2 Report

A simple proposition for the authors: the Riemann-Liouville operators should naturally be studied on H\"older spaces (cf. e.g. [Samko] Samko, S. Fractional integration and differentiation of variable order: an overview. Nonlinear Dyn 71, 653–662 (2013). https://doi.org/10.1007/s11071-012-0485-0). The proposed result is far from such considerations - why? Because one then has to investigate which space to choose, recheck the operator properties, etc. But this would bring something new to the mathematical foundations. This would be worth reading.

The most interesting (potentially) problem was written up in Remark 2 - but without any proof, and one that should be in the paper.

Perhaps another example of a style that does not suit me: Schauder's theorem (Theorem 1) is rather well known to readers and the inclusion of it is redundant (worse still, it is cited from Deimling's book, and this is the only use of this source, so it is redundant).

The problem of equivalence (or not) of differential and integral issues (cf. [Samko]) has been 'solved' by adoption by definition (Definition 4). And there was an opportunity to consider - including the very spaces under consideration on which these operators act. 

There are editorial mistakes to be corrected.

Author Response

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Author Response File: Author Response.docx

Reviewer 3 Report

See the attached file.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The answers to my objections are unsatisfactory or wrong. For example a comment about Remark 2. Instead of explaining where such equivalent conditions came from, the authors included citations of 2 papers to the presented result. The first of the proposed works does not deal with deviator equations at all and does not contain any result about the equivalence of differential and integral problems. The second citation [23] does not deal at all with the problem addressed in this note. 

Author Response

See attached file.

Author Response File: Author Response.pdf

Reviewer 3 Report

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Comments for author File: Comments.pdf

Author Response

See attached file

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

I pointed out the lack of response to the first review, and now the authors have addressed only one. The object of this paper is treated purely technically. My further comments to improve the article make no sense.

Reviewer 3 Report

All needed corrections are made.