Refinements to Relation-Theoretic Contraction Principle
Round 1
Reviewer 1 Report
As in the report!
Comments for author File: 
Comments.pdf
Author Response
|
Action:(i) Suggestion is incorporated by avoiding the logical symbols as desired.
(ii) Suggestion is incorporated by adding the required references. |
Reviewer 2 Report
In the paper under review the authors obtain a fixed point result for a generalized nonexpansive mappings which generalizes an analogous result obtained by the first and the third authors in 2015. The result is of interest for experts in the fixed point theory and the proofs are correct. But there is an issue which needs a clarification. It seems that the authors study the existence of a fixed point for mappings on a metric space with a graphs
which was studied in the seminal paper by Jachymski, Jacek The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc. 136 (2008), no. 4, 1359–1373. There are several papers which contain some further developments. It seems that the authors are not aware about these works. The authors should cite them, to compare their results with the result of the current paper and the paper of 2015.
Author Response
Reviewer #2: It seems that the authors study the existence of a fixed point for mappings on a metric space with a graphs, which was studied in the seminal paper by Jachymski, Jacek The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc. 136 (2008), no. 4, 1359–1373. There are several papers which contain some further developments. It seems that the authors are not aware about these works. The authors should cite them, to compare their results with the result of the current paper and the paper of 2015.
Action: Suggestion is incorporated by adding the relevant paragraph in Introduction section
Author Response File: Author Response.pdf
Reviewer 3 Report
The paper is in the continuous of another publication of the authors in which they present some variant of classical Banach contraction principle on a complete metric space endowed with a binary relation which, under universal relation, reduces to Banach contraction principle.
Mathematically, is correct however I suggest they explain what they did in this paper explicitly in the abstract instead of referring the readers to their previous publication. They should conclude their result by adding a short conclusion.
Author Response
Reviewer #3: I suggest they explain what they did in this paper explicitly in the abstract instead of referring the readers to their previous publication. They should conclude their result by adding a short conclusion.
Action: Suggestion is incorporated by modifying the abstract.
Author Response File: Author Response.pdf
Reviewer 4 Report
This paper contains abstract, introduction, p-self-closedness and regularity (3 known definitions), additional observations (5 known definitions, 1 new definition, 1 known theorem, 3 propositions), main results (1 new theorem, 1 example) and references (6 items).
Upon reviewing the paper I have made the following remarks:
- in whole paper it should write "+∞" instead of "∞"; "for all" instead of "∀"; "there exists" instead of "∃"; "implies" instead of "⇒"; "if and only if" instead of "⇔".
- in the proof of the Proposition 3: p(τrn,τr)≤αp(rn,r)→0 as n→+∞ (in the same line).
General opinion
This paper has been well and clearly written with a new approach. The proof of the Theorem 2 seems to be correct. This paper also contains example that support the theoretical results.
Therefore, the paper is recommended for publication after the minor revision in accordance with the above remarks.
Author Response
Reviewer #4: Upon reviewing the paper I have made the following remarks:
- in whole paper it should write "+∞" instead of "∞"; "for all" instead of "∀"; "there exists" instead of "∃"; "implies" instead of "⇒"; "if and only if" instead of "⇔".
in the proof of the Proposition 3: p(τrn,τr)≤αp(rn,r)→0 as n→+∞ (in the same line).
Action: Suggestion is incorporated by avoiding the logical symbols as desired.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
I am satisfied with the revision and recommend to publish the paper in its current form.
Author Response
Point 1: It seems that the authors study the existence of a fixed point for mappings on a metric space with a graphs, which was studied in the seminal paper by Jachymski, Jacek The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc. 136 (2008), no. 4, 1359–1373. There are several papers which contain some further developments. It seems that the authors are not aware about these works. The authors should cite them, to compare their results with the result of the current paper and the paper of 2015.
Response 1: Suggestion is incorporated by adding the relevant paragraph in Introduction section.
