The Split Equality Fixed-Point Problem and Its Applications

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors deal with a current and interesting problem, which is the computation of fixed points of maps between Hilbert spaces, which is framed within the Split Equality Fixed-Point paradigm. The approach chosen is the design of numerical methods for the computation of these fixed points. The approach is adequate and the results seem to be correct. The reading is reasonably agile (considering that the level of detail that the chosen approach requires is high, which does not help the agility of the reading). However, before being able to recommend the publication of the article in this prestigious journal, I would like the authors to review the following issues:

1. In the Introduction section, the authors provide a sufficient conceptual framework, but do not frame the topic discussed within the different and fruitful lines of research and approaches to the study of fixed points. Of particular interest is the study of fixed point problems in geometry. The authors should mention works on fixed points of automorphisms of the vector bundle moduli space over Riemann surfaces, or fixed points of principal bundles over algebraic curves.

2. Also in the Introduction, I believe that the authors should explain in greater detail the objectives of the research. Perhaps it would even be pertinent to include the statements of the main results.

3. It is a good idea to include a Preliminary section where the previous results to be used are collected. However, in its current wording, this section is obscure, because the reader does not know where these results are going. I suggest that the authors consider including these results as they are used (this is a suggestion).

4. Authors should revise the abstract. I think the first sentence is too long and I can't understand it well.

Minor comments:

- There are some typos that need to be corrected.  For example: page 17, line -3 (missing comma); page 3, line 4 after equation 1.8 (missing a space). I recommend doing a typo check.

Author Response

First of all, we thank the reviewer for very constructive suggestions. We have applied all the comments to improve the manuscript along the suggestions. The details of the comments and our responses as attached.

Thanks for your kind considerations.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Report is attached

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Author Response

First of all, we thank the reviewer for very constructive suggestions. We have applied all the comments in order to improve the manuscript along the suggestions. The details of the comments and our responses as attached:

 

Thanks for your kind considerations 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

See the attachement.

Comments for author File: Comments.pdf

Author Response

First of all, we thank the reviewer for very constructive suggestions. We have applied all the comments in order to improve the manuscript along the suggestions. The details of the comments and our responses as attached:

 

Thanks for your kind considerations 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Thanks to the authors for their review. Many of my comments have been addressed. However, I still insist that it is desirable to strengthen the introduction in relation to the first comment in my first review report. I was already aware that applications of fixed point theory to the geometric contexts mentioned therein are beyond the specific objectives of the present research. But it is to be expected from an introduction not only to present the problem, but also to justify it in relation to a variety of areas within mathematics the wider the better. I therefore suggest that the authors pay attention to my first comment in the first review report, although I recognize that the paper is, in its present version, a good one (this is only a suggestion to improve the justification of the topic).

Author Response

Reviewer’s  comment: Thanks to the authors for their review. Many of my comments have been addressed. However, I still insist that it is desirable to strengthen the introduction in relation to the first comment in my first review report. I was already aware that applications of fixed point theory to the geometric contexts mentioned therein are beyond the specific objectives of the present research. But it is to be expected from an introduction not only to present the problem, but also to justify it in relation to a variety of areas within mathematics the wider the better. I therefore suggest that the authors pay attention to my first comment in the first review report, although I recognize that the paper is, in its present version, a good one (this is only a suggestion to improve the justification of the topic).

 

Thanks for your kind suggestion and we have taken this suggestion and it has now been captured in the introductory part; please see pages 2 and 3.

 

 

Author Response File: Author Response.pdf