Coincidence Point of Edelstein Type Mappings in Fuzzy Metric Spaces and Application to the Stability of Dynamic Markets
Round 1
Reviewer 1 Report
The authors should revise the paper according to my comments and highlight the changes in new version.
Comments for author File: 
Comments.pdf
There are many typos in the text. The authors should erase them. I prepared some examples in my report for them.
Author Response
Please see the attached file which contains the response to all the (three) Reviewers.
With Thanks
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments for author File: Comments.pdf
Author Response
Please see the attached file which contains the response to all the (three) Reviewers.
With Thanks
Author Response File: Author Response.pdf
Reviewer 3 Report
This paper presents proof of a coincidence point result for a pair of mappings in fuzzy metric spaces, which satisfy an Edelstein-type contractive condition. Coincidence points are applied to describe the equilibrium of a simple demand-supply model in a dynamic market. By utilizing the coincidence point theorem in fuzzy metric spaces, the study demonstrates that a dynamic market with supply-sensitive or demand-sensitive characteristics invariably converges towards equilibrium.
Line numbers are missing.
Page 2: “In mathematical economics, the equilibrium states of a system are of particular interest. Most of the studies on economic models are subject to the analysis of the equilibrium states.” This is not necessarily true. See for example Dragicevic, A. (2019) Market Coordination Under Non-Equilibrium Dynamics, Networks and Spatial Economics, 19; Arthur, W. (2014) Complexity and the economy. Oxford University Press, Oxford.
Page 2: “If a market is not in equilibrium, then changes in variables (on which the dynamics depends) may force the market (i.e., price of commodity) towards its equilibrium, and if this is the case, then the market is called stable”. This is not necessarily true. The market can be temporarily at equilibrium without being stable. See for example Dragicevic, A. (2023) Pseudomonotone Variational Inequality in Action: Case of the French Dairy Industrial Network Dynamics, Journal of Industrial and Management Optimization; M. G. Cojocaru, P. Daniele and A. Nagurney (2005), Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications, Journal of Optimization Theory and Applications, 127.
The introductory section of the paper lacks sufficient details regarding the research problem and fails to establish clear connections between mathematical concepts and their economic interpretations within the modeling framework. Furthermore, there is a lack of explanation regarding the mapping and Jungck sequence, which are fundamental components of the study. Additionally, the absence of an outline or introduction to different sections further diminishes the clarity and organization of the paper.
Page 4: What is T1? What is GV1?
Suggesting that the “equilibrium point for a dynamic market is a coincident point of the demand function and the supply function” is interesting. You state that “the equilibrium quantity will be the corresponding coincidence point” and then you talk about the equilibrium price in Theorems 4.2. and 4.3. What is the underlying relationship between the equilibrium price and the equilibrium quantity within the context of your modeling approach?
Section 5 of your paper is deficient in terms of discussion and analysis. It would significantly enhance the quality of the section to conduct a comparative analysis, particularly by establishing parallels with the cobweb theory. By highlighting the advantages and improvements of your proposed model in comparison to the cobweb theory, you can provide a more robust justification for its superiority.
Furthermore, it is essential to elucidate the contributions of your new approach to economic theory, specifically in relation to Meznik's approach. In particular, considering the omission of differential calculus, what additional insights or advantages does your approach offer? By addressing these points, you can provide a comprehensive understanding of the novelty and significance of your research in the context of existing literature.
Author Response
Please see the attached file which contains the response to all the (three) Reviewers.
With Thanks
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
I agre with publish.
Author Response
We are thankful to Reviewer for giving his/her valuable time.
Best Regards
Reviewer 2 Report
Some suggestions are mentioned, need careful attention
Comments for author File: Comments.pdf
Author Response
Response to Reviewers:
Reviewer 1: Thanks for your consent for publication.
Reviewer 2:
- Details has been added before Definition 2.2.
- The steps before and after inequality (3) are by the definition of Jungck sequence.
- In the proof of Theorem 3.5 details has been added.
- Equality in Example 3.7 has been examined and found true.
- In the proof of Theorem 4.2, the fuzzy metric is a stationary fuzzy metric space, hence does not depends on tau.
- In section 5, we have explored theoretical possibilities for a simple demand supply model. It is worth to investigate the practical numerical example for such situations, but this is a task for the specialists from the field of Numerical Methods.
Reviewer 3: We are also thankful to reviewer.
Reviewer 3 Report
Thank you for the updates.
No comment.
Author Response
We are thankful to Reviewer for giving his/her valuable time.
Best Regards
Round 3
Reviewer 2 Report
Dear Editor
I am not agree with the reply of authors, authors must explaineach point in detail in paper or provide explanation separately, if metric is stationary, it should be mentioned in example, if it is stationary then it is very trivial example, authors must add some potential example, regarding the application, authors are consistent with the same reply as previous. Also, no updates are found before def. 2.2 as mentioned by his reply either authors uploaded the wrong file or ignored our comments.
With these changes, I cannot recommend this paper to publish. I advise the authors to carefully revise the article and upload highlighted pdf.
Best wishes
Author Response
The authors sincerely appreciate the valuable comments provided by the reviewer. However, we respectfully disagree with the reviewer's concerns regarding the perceived triviality of the examples presented in this article, specifically Example 3.2 and Example 3.7. We firmly believe that all the examples discussed in this article are indeed nontrivial and possess inherent interest. To address this concern, we kindly request the reviewer to substantiate their viewpoint with supporting evidence or specific arguments.
Regarding the application, the authors have outlined the theoretical prospects for implementing these mappings within dynamic markets to enhance market stability. Researchers who are working on numerical methods could build on the theoretical insights provided by the authors to develop numerical models that can be used to simulate the behavior of dynamic markets. These models could be used to test the effectiveness of different strategies for mitigating market instability. Regarding Definition 2.2, which pertains to the periodic points of mappings $\xi$ and $\zeta$, we acknowledge that there is a substantial body of existing literature on this topic. Consequently, we did not delve into extensive explanations. However, if the reviewer believes that further elaboration is necessary, we kindly request them to provide specific suggestions or points that they would like to see expanded upon. This would enable us to address the reviewer's concerns more effectively.In conclusion, we extend our sincere gratitude to the reviewer and the editor for their valuable contributions in improving this article. Your insights have been instrumental in enhancing the quality of our work.
Thank you,
Kind regards,
Rahul Shukla
Author Response File: Author Response.pdf
