Fixed Point and Homotopy Methods in Cone A-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation

Round 1

Reviewer 1 Report

Comments on Symmetry-1766369:

In this manuscript,  the authors introduced an ordered implicit relation in the setting of cone A-metric spaces and proved some fixed point results for some  implicit contractions via the ordered implicit relation. They also obtained the existence of Urysohn integral equation solutions and a homotopy result as an application of the main results. Some examples are provide to support the validity of the main results.

My comments on this manuscript are as follows:

1) The design and language of this paper are needed to be improved. Here are many typors and grammatical errors some of some of which are marked in red in the pdf file attached to this mail. The authors should check and correct the whole paper carefully. 

2) The manuscript is of interest for experts in the fixed point theory and nonlinear analysis.

3) The obtained results and examples are new and acceptable. In my opinion, this paper will be useful for the fixed-point theory.

    After the above comments and correcting  errors , my decision on this paper is Accept After Major Rvision.

 

Comments for author File: Comments.pdf

Author Response

1. The design and language of this paper are needed to be improved. Here are many typos and grammatical errors some of some of which are marked in red in the pdf file attached to this mail. The authors should check and correct the whole paper carefully. 

 Response:  Thank you so much for suggestions to improve the article. We have  modified the paper according to the comments highlighted on the PDF of paper.

2. The manuscript is of interest for experts in the fixed point theory and nonlinear analysis.

Response: We are grateful.

3. The obtained results and examples are new and acceptable. In my opinion, this paper will be useful for the fixed-point theory.

Response: We are grateful.

Reviewer 2 Report

The authors investigate cone metric space under the A-metric. Several fundamental theorems are presented and application to the Urysohn integral equations is explored. I think this paper is well-organized and well-written. I enjoyed reading it. I support publication and have only a few suggestions that the authors might want to consider.

1. I wasn't quite sure why the definitions of the other metrics, 2-metric, S-metric, G-metric where highlighted in section 2. The paragraph from lines 21-28 was helpful in setting up why choice of metric is important. Perhaps the author could state somewhere how the other metrics compare to the A-matric used in this paper. Maybe a few sentences in section 2 and then again in the conclusion would help the reader understand the consequences of metric choice. Alternatively, perhaps section 2 could be reduce by not including the explicit definitions of metrics not used in the rest of the paper. 

2. Line 145: I think the font on E is not correct. It should be E (math font).

3. First line on page 9 and Line number 283: The A should be bold, A

 

Author Response

1. I wasn't quite sure why the definitions of the other metrics, 2-metric, S-metric, G-metric where highlighted in section 2. The paragraph from lines 21-28 was helpful in setting up why choice of metric is important. Perhaps the author could state somewhere how the other metrics compare to the A-matric used in this paper. Maybe a few sentences in section 2 and then again in the conclusion would help the reader understand the consequences of metric choice. Alternatively, perhaps section 2 could be reduce by not including the explicit definitions of metrics not used in the rest of the paper. 

Response: Thank you so much for this suggestion, we have omitted some definitions. However, some definitions (in our opinion) are needed to maintain the story as some of their axioms have been referred in discussions.

2. Line 145: I think the font on E is not correct. It should be E (math font).

Response: We have corrected font in revised paper.

3. First line on page 9 and Line number 283: The A should be bold, A

Response: We have corrected A to be A in revised paper.

Reviewer 3 Report

The overall presentation of the paper is good but the references are not written in the same way. Some references have years of publication first and then page numbers and others have page numbers first and then years of publication. Could you go through the references and write them in the same style. 

Author Response

The overall presentation of the paper is good but the references are not written in the same way. Some references have years of publication first and then page numbers and others have page numbers first and then years of publication. Could you go through the references and write them in the same style. 

Response: We have corrected all references in the revised paper.

Reviewer 4 Report

This paper introduces and considers an ordered implicit relation that can be used to check the existence of fixed points of new contractions defined in cone A-metric spaces. The authors explore a fixed point method for proving the existence of Urysohn Integral Equation solutions. In order to obtain a homotopy result, they also establish a fixed-point theorem, where the hypothesis can be demonstrated with examples. In my opinion, the results improve, extend and develop some related recent results in the literature. This paper deserves publication and it will make a significant contribution to the literature after publication. The current version of this paper is well written and the results are quite valuable. Therefore, I am very willing to recommend the acceptation of it for publication in the Symmetry journal. However, it is worth to point out that there are some minor errors which need to be modified and corrected: 

(i)Page 6, on lines 2, 4 and 6 of the 1^st paragraph in Section 4 Main Results, “Further generalization made by”, “define some topological structure” and “that extends the idea” should be replaced by “Further generalization was made by”, “defined some topological structure” and “that extends the corresponding ones”, respectively.

    (ii)Page 8, lines 2 and 6, “also” and “consider” should be replaced by “. Also, observing that” and “we note that”, respectively.

    (iii)Page 9, on lines 4 and 6 of Remark 4.1, “Take” and “the theorems” should be replaced by “Taking” and “the corresponding theorems” , respectively.

    (iv)Page 9, on lines 1 and 3 of Theorem 4.2, “as”, “along with” and “owning” should be replaced by “is”, “with” and “owns” , respectively.

    (v)Page 10, lines 1 and 2, “Use of condition” and “also” should be replaced by “Using condition” and “. Also,” , respectively.

    (vi)Page 10, in the last sentence above Subsection 4.3, “By same reasoning” should be replaced by “Using the same reasoning”.

    In addition, during the past several decades, the fixed point theory has played a key role in solving many problems arising in nonlinear analysis and optimization theory such as differential hemivariational inequalities systems (see DOI: 10.3390/math9243173), monotone bilevel equilibrium problems (see DOI:10.1186/s13660-021-02683-y), generalized global fractional-order composite dynamical systems (see DOI: 10.23952/jnva.6.2022.1.09), generalized time-dependent hemivariational inequalities systems (see DOI: 10.3390/sym13101801) and so on. Please the authors add and insert such information in this article.

Author Response

(i)Page 6, on lines 2, 4 and 6 of the 1^st paragraph in Section 4 Main Results, “Further generalization made by”, “define some topological structure” and “that extends the idea” should be replaced by “Further generalization was made by”, “defined some topological structure” and “that extends the corresponding ones”, respectively.

Response: We have modified these sentences in the revised paper.

    (ii)Page 8, lines 2 and 6, “also” and “consider” should be replaced by “. Also, observing that” and “we note that”, respectively.

Response: We have modified these sentences in the revised paper.

    (iii)Page 9, on lines 4 and 6 of Remark 4.1, “Take” and “the theorems” should be replaced by “Taking” and “the corresponding theorems” , respectively.

Response: We have modified these sentences in the revised paper.

    (iv)Page 9, on lines 1 and 3 of Theorem 4.2, “as”, “along with” and “owning” should be replaced by “is”, “with” and “owns” , respectively.

Response: We have modified these sentences in the revised paper.

    (v)Page 10, lines 1 and 2, “Use of condition” and “also” should be replaced by “Using condition” and “. Also,” , respectively.

Response: We have modified these sentences in the revised paper.

    (vi)Page 10, in the last sentence above Subsection 4.3, “By same reasoning” should be replaced by “Using the same reasoning”.

Response: We have modified these sentences in the revised paper.

    In addition, during the past several decades, the fixed point theory has played a key role in solving many problems arising in nonlinear analysis and optimization theory such as differential hemivariational inequalities systems (see DOI: 10.3390/math9243173), monotone bilevel equilibrium problems (see DOI:10.1186/s13660-021-02683-y), generalized global fractional-order composite dynamical systems (see DOI: 10.23952/jnva.6.2022.1.09), generalized time-dependent hemivariational inequalities systems (see DOI: 10.3390/sym13101801) and so on. Please the authors add and insert such information in this article.

Response: We have added this useful information in the revised paper.

Round 2

Reviewer 1 Report

All the modifications according to the last comments are suitable. However, here are some other typors need to be corrected some of which are marked in the pdf file enclosed the mail.

Comments for author File: Comments.pdf

Author Response

Honorable Reviewer,

We have modified the paper according  your suggestions.

Thank you so much for guiding us to improve this article.