Fixed-Point Convergence of Multi-Valued Non-Expansive Mappings with Applications
Round 1
Reviewer 1 Report (Previous Reviewer 1)
Comments and Suggestions for AuthorsThis article is devoted to exploring the fixed point outcomes for multi-valued asymptotically non-expansive maps regarding convergence criteria in complete uniformly convex hyperbolic metric spaces. The famous fixed-point theorems of Goebel and Kirk along with other recent outcomes in literature can be obtained as corollaries of these main results. The adequate graph and example are also employed to bear up the conditions of the main outcomes. In addition, as applications an outcome in connection with Nash equilibrium in game theory is given to explain the value of the findings.
In my opinion, the authors have carefully absorbed and adopted the suggestions and comments given by the previous referees. This article deserves publication and it will make a meaningful contribution to the literature after publication. The current version of this article is well written and the results are quite interesting and very valuable. Therefore, I am very willing to recommend the acceptance of it for publication in the Axioms journal.
Comments on the Quality of English LanguagePlease the authors check by themself the grammar, spelling, punctuation, letters, etc.
Author Response
The authors sincerely appreciate your invaluable comments. We have diligently reviewed and rectified all grammar, spelling, punctuation, and letter errors in accordance with your advice.
Author Response File: 
Author Response.pdf
Reviewer 2 Report (New Reviewer)
Comments and Suggestions for AuthorsThe manuscript should be prepared in such a manner that the proved results are theorems or propositions or corollaries. One can guess that there are some in the text, because the authors cithe e.g.Theorem 3 or Theorem 4 , Corollary 2, but they do not exist in the paper. Thus I recommend the article should be reconsidered after such corrections. I think that it can be recommended positively then , but now it very difficult to read.
Comments on the Quality of English LanguageSome minor corrections can be done also.
E. G.;
Let ... represent ( not represents)
Similarly in other places
Author Response
Thank you for your advice and criticism. We've made the necessary changes in accordance with your directions
Author Response File: Author Response.pdf
Reviewer 3 Report (New Reviewer)
Comments and Suggestions for AuthorsThe authors present several convergence theorems for multivalued mappings. Several consequential cases are presented in detail and an application is made to game theory. The introduction does a good job of providing context for the work and a particularly good survey of the history of the area. I do support publication in Axioms but I have a couple of suggestions.
Line 56: "... when he looks that..." should read "...when he OR SHE looks AT..."
I think the use of figures is good for this paper however that needs to be improved in several ways.
i. The quality of the images needs to be improved. They are not currently of publication quality in both presentation and resolution.
ii. The captions need to be improved. The colors should be explained etc.
iii. The figures need to be discussion (or at least referenced) in the body of the work.
iv. Figure 4 is missing.
Comments on the Quality of English LanguageThe English is fine. Just a few minor issues of no consequence.
Author Response
The figures are already in the appropriate quality needed for the manuscript. The use of different colors in the figures doesn't signify any particular attribute; multiple colors have been employed solely for enhanced clarity. We have made improvements to the captions (please refer to page 3). Regarding Figure 4, it was mistakenly stated as missing earlier, but this has now been rectified.
Author Response File: Author Response.pdf
Reviewer 4 Report (New Reviewer)
Comments and Suggestions for AuthorsThe manuscript at hand is difficult to read and certainly intended only for the reader specialized to this particular field of research. I have three main points and a long list of minor ones which have to be resolved before a decision can be made whether or not the manuscript is appropriate for publication.
A. The definition of a multi-valued G-asymptotically non-expansive mapping in the beginning of Sec. 3 has to be reformatted. In moment, the title of this definition is found in square brackets in boldface inside of the running text. This is not appropriate for a definition. Therefore, I propose to change slightly by instead starting at this point a new paragraph (as for Condition etc.) with boldface "Definition [Multi-valued ... Mapping]" and after that a line break.
B. The second sentence in Sec. 5 is out of order. Together with the first sentence, this should be moved to a later place, at least after the Nash Equilibrium Theorem is mentioned. My suggestion is to move it to the end of the first paragraph. In this paragraph, there are also technical issues about the references which seem not to be defined correctly. Please resolve these issues.
C. Problems on Hadamard spaces are mentioned only in the section "Open Problem" and, therefore, cannot be mentioned in the Conclusions as a subject treated here.
Minor issues are the following:
* l.12: Instead of F.P I propose the simper acronym FP for fixed point.
* l.23: "In the year 1969" -> "In 1969"
* l.37: "[23] result" -> "result [23]"
* l.72: ". Suppose" -> ", suppose"
* l.77: "and multi-valued" -> "and the multi-valued"
* l.77f: "Then whole domain of H form" -> "Then the whole domain of H forms"
* l.80: few collections" -> "a couple of collections"
* l.86f: "named as a" -> "named a"
* l.93: "directions(see" -> "directions (see"
* l.98: "with conventional" -> "with the conventional"
* l.142: "Let D symbolizes" -> "Let D symbolize"
* l.143: "characterize" -> "characterizes"
* l.144: "loops Assuming" -> "loops, assuming"
* l.150: "with graph" -> "with a graph"
* l.156: "Let G represents" -> "Let G represent"
* l.157: "to be multi-valued" -> "to be a multi-valued"
* l.165: "in X then" -> "in X. Then", no comma after the next equation
* l.169: "of q, then" -> "of q. Then"
* l.176: "be a generalized" -> "is a generalized"
* l.180: "therefore" -> "one has"
* l.185: "so we determine" -> "we determine"
* l.190: "be a generalized" -> "is a generalized"
* l.191: " then by Condition" -> ". Then by Condition"
* l.193ff: "therefore" -> "one has"
* l.201: "be the Lipschitz" -> "is the Lipschitz"
* l.211: "therefore, we can write" -> "we can write"
* l.213: "approaching infinity and using given" -> "approach infinity and using the given"
* l.223: "of Convergence" -> "of the Convergence"
* l.226: "is assumed" -> "assumed"
* l.237: "be the directed" -> "is the directed"
* l.242: "are generalized" -> "be generalized"
* l.245: "becomes," -> "becomes"
* equation after l.250: terminated with full stop, not comma.
* l.254: "implying" -> "implies that"
* l.258: "implying (6) hold." -> "implying (6)."
* l.259: "Thus P is" -> "Thus P is a", "is an F.P" -> "is a F.P"
* l.261: "the Theorem 3's presumptions" -> "the presumptions of Theorem 3"
* l.262: ",where" -> ", where"
* l.263: "be the generalized" -> "is the generalized"
* l.265: "therefore with $b_h=1$, we" -> "with $b_h=1$ we"
* l.277f: "existence of Nash" -> "existence of the Nash"
* l.283: "space then" -> "space. Then"
* l.286: "If binary" -> "If the binary"
* l.305: "by their" -> "by its"
* l.318: "be the collection" -> "is the collection"
* l.324: "restrained" -> "realized", "uses" -> "needs"
* l.333: "investigators in the same field" -> "investigations in this field"
Author Response
Corrections done accordingly, also references are corrected
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report (New Reviewer)
Comments and Suggestions for AuthorsThe remarks are included in the file attached. Suggestion of reducing the section 5 and bibliogaphy are the main.
Comments for author File: Comments.pdf
Author Response
Dear Editor and Reviewers,
We are truly grateful for your invaluable comments and insightful feedback, which have significantly improved our manuscript, titled "FIXED POINT CONVERGENCE OF MULTI-VALUED NON-EXPANSIVE MAPPINGS WITH APPLICATIONS." We have conscientiously addressed the reviewer's comments, implemented all required changes, and transparently acknowledged these modifications. Our revised manuscript, thoughtfully aligned with your guidance, is enclosed for your review and consideration. Below are the details of our revisions:
Thank you for your continued support and guidance.
Please see the attached pdf
- The main results presented are: Theorem 3, which is a generalization of fixed point theorem by K. Goebel and Kirk [12] to some special spaces with a graph and continuous multivalued Graph asymptotically non-expansive mappings.Theorem 4 which generalizes a convergence result by Khamsi and Khan (see Corollary 1).
ACTION: NOT REQUIRED
- Example 1 is rather an illustration of the results, not an application.
ACTION: We didn’t claim that Example 1 is an application, we presented example in support of the hypothesis of the main results.
- In my opinion the application to game theory is a bit missleading. The classical Kakutani fixed point theorem and its use in the proof Nash equillibrium theorem is a classical result and has nothing to do with the results of the paper ( the mapping of best response is upper continuous but not lower continuos in general). Thus rewring them from some lecture notes [45] makes no sense. Of course they do not follow from any result of the present paper. One can mention them only. Thus Section 5 should be cut off or some real application given.
ACTION: We appreciate the feedback and the constructive suggestions provided by the reviewer. However, we respectfully disagree with the recommendation to remove Section 5 from our paper. Game theory is a field that is often described mathematically, allowing for the creation of flexible and abstract models to address a wide range of situations.In Section 5 of the paper, we used theorem3, which is rooted in the principles of Kakutani's type theorem and its connection to graph theory. This theorem is instrumental in our exploration of Nash equilibrium within the context of game theory. We believe that this unique approach not only enhances the comprehensiveness of our paper but also has the potential to pique the interest of new investigators in this field.
Furthermore, we are encouraged by the fact that 3 out of the 4 reviewers have expressed their support for retaining Section 5 in the paper. Their endorsement underscores the potential significance of this section in our work.
In light of these considerations, we kindly request that Section 5 be preserved as an integral part of our paper. We believe that it contributes significantly to the overall quality and novelty of our research, and we are confident that it will be valuable to both current and future scholars in the field of game theory.
We are open to further discussion and willing to address any concerns or questions the reviewer may have regarding Section 5. Thank you for your understanding and consideration.
- The presented bibliography can be reduced in number since the majority of the positions are cited only very generally and they are not necessary. The presentation is now quite clear and I recommend the paper to be published after the suggested reductions.
ACTION: Extra citations are removed and now only most relevant bibliography is included.
Author Response File: Author Response.pdf
Reviewer 3 Report (New Reviewer)
Comments and Suggestions for AuthorsI think the paper is a good contribution to the liturature and I support publication. There are still issues with the figures but I will defer to the editors on that.
The authors state the figures meet the standards of the journal. They do not meet the resolution standards as can be clearly seen by visual inspection. Further, the fontsize on figures 1 and 3 makes it hard to read. Figure 2 does not have it's axes labelled either on the graph or in the caption. The aspect ratio of the font in figure 2 is incorrect. The axes of figure 2 extends into figure 3.
In their response the authors state that the color is for clarity. The readers will not see this response and this is not mentioned in the paper.
The authors state that the captions have been improved. However they are the same as in the first draft except F.P is FP.
Comments on the Quality of English Language
The English is fine.
Author Response
Dear Editor and Reviewers,
We are truly grateful for your invaluable comments and insightful feedback, which have significantly improved our manuscript, titled "FIXED POINT CONVERGENCE OF MULTI-VALUED NON-EXPANSIVE MAPPINGS WITH APPLICATIONS." We have conscientiously addressed the reviewer's comments, implemented all required changes, and transparently acknowledged these modifications. Our revised manuscript, thoughtfully aligned with your guidance, is enclosed for your review and consideration. Below are the details of our revisions:
Thank you for your continued support and guidance.
please see the attached pdf file
I think the paper is a good contribution to the literature and I support publication. There are still issues with the figures but I will defer to the editors on that.
The authors state the figures meet the standards of the journal. They do not meet the resolution standards as can be clearly seen by visual inspection.
ACTION:We have improved the resolution, now figures are much more visible and readable.
Further, the fontsize on figures 1 and 3 makes it hard to read.
ACTION:Fontsize of figure 1 and 3 is now adjusted so that they are easy to read now.
Figure 2 does not have it's axes labelled either on the graph or in the caption.
ACTION:Axes of figure 2 are labeled now. All figures are of same color now and captions are improved (see page 3,4,12).
The aspect ratio of the font in figure 2 is incorrect.
ACTION: Corrected now, see page 3.
The axes of figure 2 extends into figure 3.
ACTION: Now corrected, see page 3.
In their response the authors state that the color is for clarity. The readers will not see this response and this is not mentioned in the paper.
ACTION: All figures are of same color now, colors did not represent any specific trait so any description was not provided previously.
The authors state that the captions have been improved. However they are the same as in the first draft except F.P is FP.
ACTION: Previous caption for figure 1 was “Unique FP”, now it is updated as “Graphical representation of the unique FP of the mapping defined by 1”, see page 3.
Previous caption for figure 2 was “Many FPs” now it is updated as “Visualization of the numerous FPs of the mapping defined by 2”, see page 3.
Previous caption for figure 3 was “Domain as FPs”, now it is updated as “Illustration of the infinitely many FPs of the mapping defined by 3”, see page 4.
Previous caption for figure 4 was “δ”, now it is updated as “Representation of Multi-valued δ -asymptotically non-expansive mapping defined by”, see page 12.
Author Response File: Author Response.pdf
Reviewer 4 Report (New Reviewer)
Comments and Suggestions for Authors The authors have improved their manuscript significantly by giving (necessary) structure to the text, in particular as this manuscript is aimed for a mathematically orientated journal like Axioms. In addition, the beginning of Sec. 5 is now understandable. I am happy that I could give the authors hints how to proceed, and I hope these hints will also help for forthcoming manuscripts. From my side, the manuscript is now ready for publication in Axioms.
Author Response
Dear Editor and Reviewers,
We are truly grateful for your invaluable comments and insightful feedback, which have significantly improved our manuscript, titled "FIXED POINT CONVERGENCE OF MULTI-VALUED NON-EXPANSIVE MAPPINGS WITH APPLICATIONS." We have conscientiously addressed the reviewer's comments, implemented all required changes, and transparently acknowledged these modifications. Our revised manuscript, thoughtfully aligned with your guidance, is enclosed for your review and consideration. Below are the details of our revisions:
Thank you for your continued support and guidance.
The authors have improved their manuscript significantly by giving (necessary) structure to the text, in particular as this manuscript is aimed for a mathematically orientated journal like Axioms. In addition, the beginning of Sec. 5 is now understandable. I am happy that I could give the authors hints how to proceed, and I hope these hints will also help for forthcoming manuscripts. From my side, the manuscript is now ready for publication in Axioms.
ACTION:No action is required.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis paper is devoted to studying the fixed point results for multi-valued asymptotically non-expansive maps regarding convergence criteria in complete uniformly convex hyperbolic metric spaces. The famous fixed-point theorems of Goebel and Kirk along with other recent results in literature can be obtained as corollaries of these main results. The adequate graph and example are also employed to back up the conditions of the main results. Moreover, as applications a result in connection with Nash equilibrium in game theory is given to reveal the value of the findings.
In my opinion, the outcomes are valid and the proofs are correct as well. The current version of this paper is well written and the outcomes are valuable and interesting. This paper is most closely related to the references [18, 26]. It is well known that the findings need to be strong enough to be published in the Axioms journal. Thus, there are a few comments that must be addressed before the submission can be accepted.
1) Provide a clearer explanation of the specific advantages and potential impact of the authors’ findings over existing outcomes in the literature, e.g., [18, 26].
2) Clarify the conditions under which the authors’ findings outperform existing outcomes in the literature, e.g., [18, 26].
3) Note that the conclusions in Section 6 need to be supported by the authors’ findings. So, according to the current findings, the authors should mention more concrete goal and more practical direction in this field for the future study.
Comments for author File: Comments.pdf
It is worth pointing out that there are some minor errors which need to be modified and corrected:
1)Page 1, on lines 2 and 4 of the ABSTRACT, “convergence theorems” and “among other recent results” should be replaced by “convergence criteria” and “along with other recent results”, respectively.
2)Page 2, on lines 3-4 of paragraph 1 of Section 2, “be” and “multi-valued valued” should be replaced by “are” and “multi-valued”, respectively.
3)Page 2, on line 2 of paragraph 4 of Section 2, “ ” should be replaced by “ ”.
4)Page 4, lines 2-3 of Definition 2, “generated as and” and “generated by ” should be replaced by “generated from by” and “generated from ”, respectively.
5)Page 4, on line 5 of the final paragraph, “for ” and “will be unique” should be replaced by “for some ” and “is unique”, respectively.
6)Page 4, on line 8 of the fine paragraph, “whenever a, b, x, y are the members of X, and χ belongs to [0,1],” should be replaced by “for all and ” because the original expression does not reveal the arbitrariness of these elements.
7)Page 5, on item (2) of Theorem 1, “Let Z∈CC(X), then any type function η defined from X to [0,∞)” should be replaced by “If Z∈CC(X), then any type function η: X to [0,∞)”.
Author Response
We want to extend our deep appreciation to the Reviewer for their invaluable comments and constructive
feedback, which have greatly enhanced the quality of this manuscript. A revised version of the paper: “Fixed
Point Convergence of Multi-Valued Non-Expansive Mappings with Applications” is attached. We have
revised the manuscript in light of the reviewer’s comments and made all required changes along with the
details that how and where the comments have been acknowledged. We wish to submit revision of the article
according to your advice.
Reviewer 1’s comments:
[1] Provide a clearer explanation of the specific advantages and potential impact of the
authors’ findings over existing outcomes in the literature, e.g., [18, 26].
[2] Clarify the conditions under which the authors’ findings outperform existing outcomes in
the literature, e.g., [18, 26].
Action: We appreciate the reviewer’s feedback and have carefully considered their comments. Our findings
align more closely with the results presented in references [2] and [18]. However, our contributions extend
beyond these references, as our results cover a broader scope by incorporating the graphic structure of the
space. Notably, the main results in above mentioned references emerge as consequences of our established
results. (Section 4, Line# 1, 2 and Remark 3)
[3] Note that the conclusions in Section 6 need to be supported by the authors’ findings. So,
according to the current findings, the authors should mention more concrete goal and more
practical direction in this field for the future study.
Action: In response to the reviewer's feedback, we have made the conclusion section broader to include more
information about our findings. Also, we have added a clear and strong open problem that points the way for
future research. (Section 6)
[4] In addition, it is worth pointing out that there are some minor errors which need to be
modified and corrected.
Action: We have thoroughly reviewed and fixed all the errors that were mentioned
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsMy remarks where pointed out in the manuscript (review_manuscript.pdf) and explained iat the end of the document.
Comments for author File: Comments.pdf
Author Response
We would like to express our sincere gratitude to the Reviewer for the invaluable comments and constructive
feedback, which have significantly contributed to the improvement of this manuscript. A revised version of
the paper: “Fixed Point Convergence of Multi-Valued Non-Expansive Mappings with Applications” is
attached. We have revised the manuscript in light of the reviewer’s comments and made all required changes
along with the details that how and where the comments have been acknowledged. We wish to submit
revision of the article according to your advice.
Reviewer 2’s comments:
[16] The Definition 5, in the way that it is formulated, is not quite a definition because it is
not very clear what is property (R) and who has it. In fact the definition is a part of a notion
introduced in [42] at page 4038, line 11. So, it has to be reformulated. For example: We say
that a CUCHMS X has the property (R) if for any decreasing sequence.....But, here is another
problem. Working with CUC you cannot define the property because any CUC has this
property (see [42], Theorem 2.2.). So you have to use exactly the definition from [42].
Action: We have followed your suggestion and incorporated this property into the statement of Theorem 1 as
you recommended.
[17] Assertions (2) and (3) are not in [42] at least not in the way that are formulated here.
Action: Thank you for your comment. In the previous version, we mistakenly referred Theorem 1 to the
wrong source, [42]. We have fixed it in the revised paper, aligning with reference [18], where these
assertions are stated similarly to how we've presented them in our paper.
[4] I am not sure how condition (A) implies the conclusion.
Action: In response to your comment, we have elaborated the steps in the proof of Theorem 3 to enhance
clarity and understanding, particularly regarding how condition (A) leads to the conclusion.
Furthermore, we have made all the changes you suggested and corrected all the errors you pointed out in rest
of the comments
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsIn our opinion, the present paper is not publishable. For more details, please refer to the attached PDF file.
Comments for author File: Comments.pdf
Extensive editing of English language required.
Author Response
Revision Note
Article ID
Title: Fixed Point Convergence of Multi-Valued Non-Expansive Mappings with Applications
Dear Reviewer,
Our sincere thanks go to the Reviewer for the invaluable comments and constructive feedback, which
played a vital role in enhancing the quality of this manuscript. A revised version of the paper: “Fixed
Point Convergence of Multi-Valued Non-Expansive Mappings with Applications” is attached. We have
revised the manuscript in light of the reviewer’s comments and made all required changes along with the
details that how and where the comments have been acknowledged. We wish to submit revision of the
article according to your advice.
Reply to Reviewer -3
This Reviewer pointed out some ridiculous mistakes at two places in the text. In fact, these are dull
typographical errors, and we are thankful to the reviewer for pointing this out.
ACTION:
These errors have been removed, the following text has been added, and similar relevant previous text has been
removed.
For two sets
X,Y , a multivalued mapping is a set valued function from
X
to
2 ,
Y
the
power set of
Y
. Consider a function
such that is a square root of .
Then
T
is a multi-valued mapping with
T(0) {0}
and
T(1) {1,1} . Here it should be
noted that
1T(1)
and
0T(0)
, that is,
T
has F.Ps.
Moreover, the proofs of all theorems have been inspected again carefully
Author Response File: Author Response.pdf
