On Modified Interpolative Almost
Round 1
Reviewer 1 Report
The paper defines a new structure of a generalized metric space, which provides an extension of both a partial modular metric space 2 through a b-metric space and explains the topological aspects of new examples of space realizations. This work is of undoubted interest and I recommend it for publication in the journal Axioms.
Author Response
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Author Response File: 
Author Response.pdf
Reviewer 2 Report
The work under review is devoted to the existence and uniqueness of fixed points. This topic is almost inexhaustible. There is a huge number of works in one way or another related to this topic. And this is despite the fact that, in fact, we are talking about two theorems: the metric and the ordinal version. This article generalizes theorems of metric content, the idea of which goes back to S. Banach. From the advent of the classical theorem to the present, theorems have been proved both directly generalizing it and theorems with various modifications of the metric function. In this paper, 18 properties of the metric function and the family of metric functions are listed. The main results are proved for a function that satisfies four or five of the above properties. I note that all results are provided with sufficiently detailed proofs. However, they are difficult to read, because the designations chosen by the authors are too cumbersome and seem far-fetched. From the point of view of the reviewer, the use of Gothic and other beautiful letters does not improve the perception of the text. The same applies to three-story designations. After all, in Definition 1 the authors did without them. It is clear that you want to reflect as much as possible in the designation, but you also need to take care of the reader. In addition, the conditions imposed on metrics often seem far-fetched. Although the examples given make it clear. On the other hand, the examples given on pages 5 and 6 give the key to the description of all families of metric functions on the line. It would be interesting to highlight this. In general, I think that the work is of academic interest and should be published. Although, most likely, it should have been submitted to the journal "Mathematics".
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
In this paper the authors introduce the notion of partial modular b-metric space that extends the known generalizations of metric spaces. They showed that the introduced notion strictly extends the notions of partial modular spaces and modular b-metric spaces. They prove a fixed point theorem for an interpolative type contraction in a complete partial modular b-metric space and give its application to homotopy theory.
The paper is clearly organized. It contains a comprehensive introduction describing various generalizations of metric spaces. The proofs of properties of partial modular b-metric spaces and the fixed point theorem are clearly presented. The examples illustrating the discussed properties are provided with detailed explanations.
Author Response
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Author Response File: Author Response.pdf
Reviewer 4 Report
The article is completely abstract in nature and, apparently, the results do not imply any applications. In addition, apparently, some systems were used to translate original material into English. As a result, the article contains fragments containing text that is not used in mathematical literature.
I believe that it is necessary to give the motivation that prompted the authors to introduce new generalizations of metric spaces, see section. 2. It is necessary to make a comparison with the known concepts specified, for example, in definition 1. The main thing that needs to be done is to indicate the advantages that the new concepts introduced by the authors give. Otherwise, the article looks like another format, rather complex, but not very interesting collection of technical calculations.
Author Response
"Please see the attachment."
Author Response File: Author Response.pdf
Reviewer 5 Report
Authors introduce partial modular b-metric, which is more general than partial modular metric space. Then, they prove an Interpolative type contraction mapping theorem. On page 2 I am confused about Definition 1 or the consequences of the the space, then axioms, these are not the axioms. The main result is Theorem 3 and from this deduction of several earlier results. Although I could not check all numbers in Example 4, it illustrates the importance of the obtained result. Theorem 4 has to be supported by an example. Throughout the paper, English has to be revised, currently it is rather bad. The word Novel has been used 9 times in the paper, what is the novelty. The paper gives some extensions of the known results.
Authors introduce partial modular b-metric, which is more general than partial modular metric space. Then, they prove an Interpolative type contraction mapping theorem. On page 2 I am confused about Definition 1 or the consequences of the the space, then axioms, these are not the axioms. The main result is Theorem 3 and from this deduction of several earlier results. Although I could not check all numbers in Example 4, it illustrates the importance of the obtained result. Theorem 4 has to be supported by an example. Throughout the paper, English has to be revised, currently it is rather bad. The word Novel has been used 9 times in the paper, what is the novelty. The paper gives some extensions of the known results.
Author Response
"Please see the attachment."
Author Response File: Author Response.pdf
Round 2
Reviewer 4 Report
The authors have made some corrections to the text, which as a whole has become somewhat better.
From my point of view, it would be advisable to give possible applications of the results obtained in the article. Unfortunately, this has not been done, however, it may represent a subject of further research.
Reviewer 5 Report
I am satisfied with the revision.
