Generalized Contractions and Fixed Point Results in Spaces with Altering Metrics

Round 1

Reviewer 1 Report

 

The article deals with some fixed point results for mapping satisfying come certain contractive conditions. The manuscript contains three sections plus the abstract and references. The manuscript is worth consideration and very well written, however, before accepting this manuscript for publication, the following remarks and points are to be addressed:

 

1)      The novelty of the manuscript is not to a high standard. The authors have just considered an upper semicontinuous function instead of the function used in Reference [11] (which is listed in the references in the manuscript) in order to introduce their new results using the same type of contraction that was used in Reference [11], so the authors are required to explain and demonstrate clearly how one would consider this as new novel work but not just a mere adaptation of the results in [11].

2)      The English looks good, however, there are few grammar or punctuation typos that need to be corrected. I shall mention some mistakes and typos below in order to just list some but there others that the authors need to correct.

a)      Page 1, Line 12 and 14 from bottom: there should be a comma after the equations

b)      Page 2, Line 1 from Top: on à in

c)      Page 3, Line 11 from Top: add “is” after “X” and before “an operator”

d)      Page 3, Line 4 from bottom: add “is” after “X” and before “an operator”

e)      Page 7, below Line 135: delete one of the full stop there.

f)       Page 8, Line 159: add a full stop at the end

 

3)      Authors need to improve the manuscript by giving at least one application where one can see the usefulness of their result and where one can apply these results.

4)      Page 2: I really do not understand why the section name is “Materials and Methods”, can’t the authors give a proper name for such section?

5)      Page 3: The authors listed directly Theorems 6, 7, and 8, however, it will be better if the authors say or mention the reason why these theorems are stated here, e.g., the following Theorems will be used in order to prove…. Etc.

6)      Page 3: Theorem 5 and Theorem 9 are the same, why is the repletion? Need to delete one of them and in addition the text before the Theorem as it is repeated as well.

 

7)      The literature is very poor and many more recent work that uses altering distances is not mentioned. Therefore, the literature needs to be improved and at least some of the recent work that uses altering distances needs to be added such as the work below:

 

            a)  doi: 10.3934/math.2020284

b)      http://jaem.isikun.edu.tr/web/index.php/archive/114-vol12no1/808

c)      DOI:10.31801/cfsuasmas.970219

Many other recent references can also be added.

 

Once these remarks are taken into consideration, the paper may be accepted for publication in the journal.

 

Author Response

Dear reviewer,

first of all we would like to thank you for your valuable sugestions and to inform you that we have considerd them us follow:

1) We have built Example 2 in order to give an example of function \phi which is not satisfying the setup from reference A.N. Branga, I.M Olaru, Some Fixed Point Results in Spaces with Perturbed Metrics  Carpathian Journal of Mathematics, 2022, 38(3), 641--654. Moreover we have added Example 3 as an application of our results. We went further and tried to extend the results from other references and consequently we gave Lemma 3 and Theorem 8 where we  considered multiple arguments (d(x,y), d(x,T(x)), d(y,T(y)), d(x,T(y)), d(y,T(x))) for function \varphi . Therefore we  showed (see Corollaries 1 &2)  that the results from Khan, M.S., Swalesh, M. and Sessa, S., Fixed point theorems by altering distances between the points, Bull. Australian Math. Soc., 1984, 30, 323--326 are obtained as  particular cases of Theorem 8.

2.  We have corrected the English topics mentioned by you 

3. We have renamed the section“Materials and Methods” by "Preliminaries" and based on your sugestion I have removed from there all duplicate and useless statements(e.g Theorem 5 and Theorem 9). Also we have metioned where the results from preliminaries are used

4. We have added the references [11], [12] and [13]

 

Reviewer 2 Report

The results presented in the manuscript are interesting. However, the author should address some matters. I suggest major revision of the manuscript based on the review report. Find the attachment in the system.

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

first of all we would like to thank you for your valuable sugestions and to inform you that I have considerd them us follow:

1) We have built Example 2 in order to give an example of function \phi which is not satisfying the setup from reference A.N. Branga, I.M Olaru, Some Fixed Point Results in Spaces with Perturbed Metrics  Carpathian Journal of Mathematics, 2022, 38(3), 641--654. Moreover we have added Example 3 as an application of our results. We went further and tried to extend the results from other references and consequently we gave Lemma 3 and Theorem 8 where we  considered multiple arguments (d(x,y), d(x,T(x)), d(y,T(y)), d(x,T(y)), d(y,T(x)))for function \varphi . Therefore we  showed (see Corollaries 1 ,2)  that the results from Khan, M.S., Swalesh, M. and Sessa, S., Fixed point theorems by altering distances between the points, Bull. Australian Math. Soc., 1984, 30, 323--326 are obtained as particular cases of Theorem 8

2. We have adresed also your minor comments from review report

Reviewer 3 Report

The comments can be seen in the attached file.

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

first of all we would like to thank you for your valuable sugestions and to inform you that I have considerd them us follow:

1) We have built Example 2 in order to give an example of function \phi which is not satisfying the setup from reference A.N. Branga, I.M Olaru, Some Fixed Point Results in Spaces with Perturbed Metrics  Carpathian Journal of Mathematics, 2022, 38(3), 641--654. Moreover we have added Example 3 as an application of our results. We went further and tried to extend the results from other references and consequently we gave Lemma 3 and Theorem 8 where we  considered multiple arguments (d(x,y), d(x,T(x)), d(y,T(y)), d(x,T(y)), d(y,T(x)))for function \varphi . Therefore we  showed (see Corollaries 1 ,2)  that the results from Khan, M.S., Swalesh, M. and Sessa, S., Fixed point theorems by altering distances between the points, Bull. Australian Math. Soc., 1984, 30, 323--326 are obtained as  particular cases of Theorem 8.

2.  We have corrected the English related topics

3. We have added explanations related to a,b,c from Introduction section

4) We have deleted brackets around \forall

5) we have done the statement of Theorem 1 more clear

6) we have explained who is F_U line 143 in manuscript

7. we have explained why \psi\circle d is not a metric

8, We compared(see Corrollarry 1 and 2)  the proposed results with the results introduced earlier .

9. We have pointed to the papers recommended  by you in regards with recent applications of fixed point theory

Reviewer 4 Report

The authors  established some fixed point results for a self mapping satisfying a newly introduced generalized contractive condition on an alternating metric space. The results obtained by the authors are correct, interesting and very good. However, the authors need to carryout the following:

1. In example one, number 2,  the authors should show that $\phi \circ d$ is not a metric on X. 

2. The authors should take time to check for the correct punctuations after each equations. 

3. If possible, the authors should apply the obtained result to delay differential equation, integral equation etc. 

4. The authors established existence and uniqueness results. Is it possible to give a numerical example  that establish this theoretical fact.

Author Response

Dear reviewer,

first of all we would like to thank you for your valuable sugestions and to inform you that I have considerd them us follow:

1) We have built Example 2 in order to give an example of function \phi which is not satisfying the setup from reference A.N. Branga, I.M Olaru, Some Fixed Point Results in Spaces with Perturbed Metrics  Carpathian Journal of Mathematics, 2022, 38(3), 641--654. Moreover we have added Example 3 as an application of our results. We went further and tried to extend the results from other references and consequently we gave Lemma 3 and Theorem 8 where we  considered multiple arguments (d(x,y), d(x,T(x)), d(y,T(y)), d(x,T(y)), d(y,T(x)))for function \varphi . Therefore we  showed (see Corollaries 1 ,2)  that the results from Khan, M.S., Swalesh, M. and Sessa, S., Fixed point theorems by altering distances between the points, Bull. Australian Math. Soc., 1984, 30, 323--326  are obtained as  particular cases of Theorem 8.

2.  We have corrected the English related topics and punctuations after each equation

3. We have explained why \psi\circle d is not a metric

 

Round 2

Reviewer 1 Report

The authors have adequately responded to my concerns in the revised manuscript. However, there are still few typos that need to be checked and corrected.

For instance:

1) after equation (9) there should be a comma or "and" and after equation (10) need to add ", respectively".

2) Also, page number need to be corrected, for instance after page 6 of 6, we start again with 0 of 6.

3) Moreover, in particular in page 0 of 6, first line from the top, at the end, either there will be a full stop or change "Then" to be "then".

4) In corollary 2, the numbering of the points need to be corrected, or there is something missing there.

5) In example 3, there is still some punctuation issues, like missing commas, see "respectively Example 2"

Please make sure that such things are done and corrected properly.

 

 

Author Response

Dear reviewer

in regards with your latest suggestions, we considered them and updated the manuscript by highlighting the changes. 

Best regards,

Marian Olaru and Adrian Branga 

Reviewer 2 Report

The results in this paper are very interesting and valuable in this area. I found unsuitable results in the manuscript. I recommend the paper for publication in Mathematics after major revising the comments. Please find the review report via the system.

 

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

first of all we would like to thank you for your valuable suggestions. We inform you that your comments were adressed as follows:

1) we detailed  R_+ from page 1

2) we revisited the Definition 2 in order to clarify your comment related to k

3) we built an additional example(Example 4) as an application of Theorem 8.

Best regards,

Marian Olaru and Adrian Branga.

Reviewer 3 Report

The paper can be sent for publication.

Author Response

Dear reviewer 

we would like to thank you for your valuable suggestions

Best regards,

Marian Olaru and Adrian Branga

Round 3

Reviewer 2 Report

In this version, the comments have been improved. This revised version can be published in the present form.