The First Rational Type Revised Fuzzy-Contractions in Revised Fuzzy Metric Spaces with an Applications

Round 1

Reviewer 1 Report

I recommend that the authors should checked and revised the manuscript carefully since

1. There are many typos and grammatical errors, also, there are many of punctuation errors.

 2. Abstract needs to improve by showing the idea of rational type revised fuzzy contraction mappings in revised fuzzy metric spaces. 

3. Introduction need improved by adding latest study in the literature.

4. The definition of $G$-complete  RFM spaces and $G$-Cauchy should be mentioned in the presentation.

5. the manuscript need to be reorganized

Comments for author File: Comments.pdf

Author Response

The following Comments and suggestions have been rectified.

1. The grammatical & punctuation errors has been corrected in the manuscript.
2. Abstract has been improved.
3. Introduction section has been improved and added in litrature.
4.  G-complete RFM-space, G-complete, G-cauchy has been mentioned in Definition 2.8, Definition 2.12, Definition 2.13.
5. The manuscript has been re-organized 

The corrections is highlighted in yellow colour background

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper presents some new interesting results on fixed points in revised fuzzy metric spaces.  However, it is written quite badly and the major revision is necessary. My remarks are following:

The authors should mention in the introduction about connections between t-norm and t-conorm. It seems that results for Revised fuzzy metric spaces can be derived from analogous results for fuzzy metric spaces. Is it true? If yes, then it would be good to explain shortly why to use the notion of Revised fuzzy metric spaces.

In Definition 2.3: what is "Revised fuzzy set on \oplus"? Is it defined by conditions 1)-5)? It is unclear.

Definition 2.4: what is p_j? What is G for?

In Definition 2.5 "there is" should be probably deleted.

I do not understand Lemma 2.7; maybe "\Leftrightarrow" should be replaced by ". Then".

In (5) "\exists" should be probably deleted.

In Theorem 3.2: What is $\mathbb{G}$-complete space? "called" should be deleted.

The paper is full of mistakes similar to those listed above. They should be carefully corrected.

Author Response

The following Comments and suggestions have been rectified.

1. RGV-fuzzy metrics are equivalent to GV-fuzzy metrics, the theories based on these concepts are equivalent. The difference is in the definitions, the proofs, and the interpretations of results. In particular, in case of revised fuzzy metrics, we have the natural interpretation of the standard situation: the longer the segments of two infinite words taken into consideration, the more precise the obtained information about the closeness of the two words.

2. It is well-known that GV-fuzzy metrics are non-decreasing in the third variable. From here, or independently, by analyzing the definition of an RGV-fuzzy metric, we conclude that RGV-fuzzy metrics are non-increasing in the third variable. This allows us to give the following visual interpretation of an RGV-fuzzy metric. Assume that we are looking from a distance  at a plane filled up with pixels. We estimate the distance between pixels x and y by means of an RGV-fuzzy metric . Being close to the plane, we see quite clearly how far the two pixels x and y are. However, going further from the plane, our ability to distinguish the real distance between different pixels becomes weaker and, at some moment, two different pixels can merge into one in our eye-pupil.
3. G-complete RFM-space, G-complete, G-cauchy has been mentioned in Definition 2.8, Definition 2.12, Definition 2.13.
4. There is has been deleted.
5. The grammatical & punctuation errors has been corrected in the manuscript.
6. Abstract has been improved.
7. Introduction section has been improved and added in litrature.

The corrections is highlighted in yellow colour background

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

It is my recommendation that this manuscript be accepted for publication in your journal.

Reviewer 2 Report

The paper has been improved and I think that it can be published now after a suitable minor correction of English.