Hybrid Fixed Point Theorems of Fuzzy Soft Set-Valued Maps with Applications in Integral Inclusions and Decision Making

Round 1

Reviewer 1 Report

Dear Authors,

 

After reading your manuscript entitled Hybrid fixed point theorems of fuzzy soft set-valued maps with applications in integral inclusions and decision making I found just some minor corrections. In the following you will find them enumerated.

 

Please, check the punctuation in whole article (i.e. before point and comma delete the empty space.)

Please, resolve the alignment problems on the following pages in the following equations:

pag. 11, before equation 4.5,

pag 13, equation 4.11

pag 14, equation 4.12

pag. 16, equation 4.17

pag 17, Corollary 4.7, in equation

pag. 18, Corollary 4.8, in the system of equations

pag 19, H3, in equation

On page 16:        9=\sigma(2,5) > \sigma(2,4)+\sigma(4,5)=4+1=5.

 

I found illustrative your both examples. For this reason, I suggest that you could write few sentences about application of your method, giving concrete possible applications in some particular field of science to emphasize your methods’ significance for readers.

In addition, I suggest comparing the roles of b-hybrid fuzzy soft set-valued maps with other maps in decision making problems, especially applicable in everyday life.

 

I enjoyed reading your article. This is well analyzed and well written manuscript describing the b-hybrid fuzzy soft contraction in b-metric spaces and the existence of fuzzy soft fixed points for such mappings.

 

Could you tell me more about the benefit to use fuzzy soft set-valued maps instead of fuzzy soft set or soft set?

 

Author Response

 

 

Comments from Reviewer 1:

Please, check the punctuation in whole article (i.e. before point and comma delete the empty space.)

Please, resolve the alignment problems on the following pages in the following equations:

pag. 11, before equation 4.5,

pag 13, equation 4.11

pag 14, equation 4.12

pag. 16, equation 4.17

pag 17, Corollary 4.7, in equation

pag. 18, Corollary 4.8, in the system of equations

pag 19, H3, in equation

On page 16:        9=\sigma(2,5) > \sigma(2,4)+\sigma(4,5)=4+1=5.

 

I found illustrative your both examples. For this reason, I suggest that you could write few sentences about application of your method, giving concrete possible applications in some particular field of science to emphasize your methods’ significance for readers.

In addition, I suggest comparing the roles of b-hybrid fuzzy soft set-valued maps with other maps in decision making problems, especially applicable in everyday life.

 

I enjoyed reading your article. This is well analyzed and well written manuscript describing the b-hybrid fuzzy soft contraction in b-metric spaces and the existence of fuzzy soft fixed points for such mappings.

 

Could you tell me more about the benefit to use fuzzy soft set-valued maps instead of fuzzy soft set or soft set?

 

 

Here is a point-by-point response to the reviewers’ comments and concerns.

 

Response to Reviewer 1:

• The whole manuscript has been double-checked and all typographical/punctuation errors have been fixed.
• The equations’ alignment problems on Pages 11, 13, 14, 16, 17, 18 and 19 have been resolved accordingly.
• The typographical errors on Page 16 have been fixed.
• Regarding the benefit to use fuzzy soft set-valued maps instead of fuzzy soft set or soft set, kindly see Page 2, Paragraphs 3-4, and Figure 4 in Page 2.

•  

  We look forward to hearing from you in due time regarding our submission and to respond to any further questions and comments you may have.

   

  Sincerely,

  Akbar Azam (On behalf of all co-authors).

Reviewer 2 Report

Please see the attachment

Comments for author File: Comments.pdf

Author Response

please see attachments. 

Author Response File: Author Response.docx

Reviewer 3 Report

This paper contains abstract, introduction (the motivation for research), preliminaries (9 known and 2 new definitions, 2 remarks, 2 known lemmas), decision making using fuzzy soft set-valued maps (1 new example, 2 new tables, 3 new figures), hybrid fixed point theorems of fuzzy soft set-valued maps (1 new lemma, 1 new definition, 1 new theorem, 1 new example and 5 new corollaries), solvability of Fredholm-type Integral Inclusions via fuzzy soft set-valued maps (1 significant new result), conclusion (the obtained results) and references (23 items).

Upon reviewing the paper I have made the following remarks:

- in whole paper it should write "+∞"  instead of "∞";

- "lim_n→+∞" instead of  "lim_n→∞";

- "∑^+∞" instead of "∑^∞";

- Check the names of all the journals in the list of references

including the abbreviated of ones;

- the list of references should include the following 2 items:

1. Suzana Aleksić et al., Picard sequences in b-metric spaces, Fixed Point Theory 21 (2020), No. 1, 35-46, DOI: 10.24193/fpt-ro.

2.  P. Debnath et al., Metric Fixed Point Theory, Applications in Science, Engineering and Behavioural Sciences, Springer 2021,

which are important for this field of study.

Otherwise, this article introduced the concept of b-hybrid fuzzy soft contraction in b-metric spaces and investigated conditions for the existence of fuzzy soft fixed points for such mappings. The preeminence of the new contractive inequalities and the associated invariant point results is that several existing results can be easy deduced as consequences. An illustrative example is provided to indicate one of the roles of fuzzy soft set-valued maps in decision making problems. As a further application, an existence result for a Fredholm-type integral inclusion is provided via the notion of fuzzy soft set-valued maps. Within the realms of differential equations and fixed point theory of both crisp and non-classical sets, it is observed that the notions put forward in this work unify and extend a significant number of developments in the related literature.

General opinion 

In this article, the authors studied a new contraction and proved some fixed point theorems in fuzzy soft set-valued maps with applications in integral inclusions and decision making. All the proofs seem to be correct. This paper also contains examples which support the theoretical results. 

 

 

 

Author Response

Journal: Mathematics

Manuscript ID: mathematics-2155731

 

Title of Paper: Hybrid fixed point theorems of fuzzy soft set-valued mapswith applications in integral inclusions and decision making

 

 

 

 

Comments from Reviewer 2:

Upon reviewing the paper I have made the following remarks:

- in whole paper it should write "+∞"  instead of "∞";

- "lim_n→+∞" instead of  "lim_n→∞";

- "∑^+∞" instead of "∑^∞";

- Check the names of all the journals in the list of references

including the abbreviated of ones;

- the list of references should include the following 2 items:

1. Suzana Aleksić et al., Picard sequences in b-metric spaces, Fixed Point Theory 21 (2020), No. 1, 35-46, DOI: 10.24193/fpt-ro.
2. P. Debnath et al., Metric Fixed Point Theory, Applications in Science, Engineering and Behavioural Sciences, Springer 2021,

 

 

 

Here is a point-by-point response to the reviewers’ comments and concerns.

 

Response to Reviewer 2:

• We have replaced "+∞"  with "∞"; "lim_n→+∞" with  "lim_n→∞" throughout the manuscript.
• The lists in the reference have all been reformatted appropriately.
• The suggested related articles have been consulted and duly cited.

 

We look forward to hearing from you in due time regarding our submission and to respond to any further questions and comments you may have.

 

Sincerely,

AKBAR AZAM  (On behalf of all co-authors).

 

Reviewer 4 Report

All is as in the report!

Comments for author File: Comments.pdf

Author Response

Reviewer 3 comments:

Remarks:
In whole paper write:
• +∞ instead only ∞;
+

∞
n=0
instead
∞
n=0
it is nicer and clearly, +∞ instead
only ∞ because what means ∞ without definition?
Also, limn→+∞ instead as in text.
etc,...
The reviewer suggests adding the next recently item:
1. Suzana Aleksi´c, Zoran D. Mitrovi´c and Stojan Radenovi´c, Picard
sequences in b-metric spaces, Fixed Point Theory 21 (2020), No. 1, 35-46,
DOI: 10.24193/fpt-ro. 

Response to Reviewer 3:

• We have replaced "+∞"  with "∞"; "lim_n→+∞" with  "lim_n→∞" throughout the manuscript.
• The suggested related articles have been consulted and duly cited.

We look forward to hearing from you in due time regarding our submission and to respond to any further questions and comments you may have.

 

Sincerely,

Akbar Azam (On behalf of all co-authors).

Round 2

Reviewer 2 Report

Dear respected authors,

here are my comments regarding the revised version:

1. As a general comment, let me suggest that the tendency to avoid the similarity percentage degrades the readability of the paper.

2. The "consistence" with reference [5] is still unclear. I my opinion the precise result(s) from the reference [5] have to be mentioned, otherwise the reader has to guess what is exactly meant by "consistent with [5]...".

3. Even if the assertions of Lemma 4 can be easily checked, in my opinion, this checking has to contained within the paper as this is the key lemma. Moreover, again in my opinion, would be more important than an example which is only indirectly  related to the main notion of the paper.

4. Just above Lemma 4.1 the authors define "a distance function" $\varsigma_{E}^{\infty}$. The proof of the fact that this is indeed a distance function seems to be lacking. Moreover, apparently, it is not used in the text.

5. A conceptual remark. The example used in order to illustrate a more sophisticated theory has to be such that the already known techniques would fail. Please note that by this I do not mean an alternative approach. In particular, the existence of solutions for the problem (5.1) can be shown using simpler theories.

Kind regards.

Author Response

please see attachments 

Author Response File: Author Response.docx

Round 3

Reviewer 2 Report

Agree