Method for Approximating Solutions to Equilibrium Problems and Fixed-Point Problems without Some Condition Using Extragradient Algorithm
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsDear Authors,
The paper is well written with a nice introduction to the topic.
The Theorems, corollaries and examples are well written.
The only issue for me is the language, there are so many errors that it is hard to read. The whole paper MUST be revised for the language as it is in a very bad condition now.
For example, "For solving the equilibrium problem, many authors have been assumed that the bifunction f : C × C → R satisfying" this is wrong.
Also, the declarations are not supposed to be at the end but before the bibliography, that must be changed too.
It would be nice if you could also cite more relevant research, from 2024 if possible.
Overall the paper is good but must be revised due to the problems I mentioned.
Revise according to the above remarks.
Author Response
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Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsDear Editor
In the present paper, an iterative problem is considered to be solved by fixed point theorems. An application is considered, too. It seems that the paper has new issues to be published but I couldn't find a new, extraordinary method for this paper as well as the application isn't novel and attractive. So I don't recommend this paper for publishing but it can be re-reviewed by considering the following major issues:
1- Similarity: The authors applied many parts of the other papers without any changing that it's against the ethical codes for to publish at MDPI.
2- Scientific issues: When the authors use of iterative algorithm a comparison is inevitable. Especially in Example 2.
3- References: They aren't updated as there is a big gap among recent works and this paper in term of time. Also, there are too references but a few number of them are used in preliminaries and as a result, some unnecessary citations should be removed.
4- Quality of writing: It's not good to use mathematical style in the text, especially, for the expersion "for all", "there exists", "so/such/provided that" etc. Moreover, some claer parts should be removed like standard definitions in preliminaries or simple computation in main theorem (Theorem 1-pages 7-11). Notice that there are many errors for using "the", "a", "an" etc.
5- Novelty: I couldn't find the role of Theorem 2 and Corollary 1 as an application. What thing Theroem 2 can improve compared to former results. To be honest, these type of papers can't be served as a new method if they don't get good views. They can be determined by strong examples not routine examples to show the validity of a new proved theorem.
After making major changes mentioned above I will be able to assess the paper better.
Sincerely yours,
A referee
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsIn this paper, the authors propose a new method for the extragradient algorithm, aiming to identify a common element of the set of fixed points of nonexpansive mappings and the set of solutions of equilibrium problems for a pseudomonotone Lipschitz-type continuous bifunction.
The results seem correct.
However, I have a few comments.
Page 1:
Line 7: add one space before (
In the end of the first paragraph of Introduction, add one space before [
In line 8 from Introduction section: ``satisfying`` should be ``satisfies``
Line 13: ``See, ...`` should be on a new line
Line 3 from the bottom: ``, the equilibrium`` instead of ``, then the equilibrium``
Line 2 from the bottom: add one space before (
Page 2:
Line 8 from top: add one space before [
Lines 16 and 17: Section or Sect.? (all over this paragragh)
Line 19: `` The`` should be ``the``
Line 20: add one space before (
In line 6 of Preliminaries section: ``given`` should be ``is given``
Stop the \it mode for the text after ``QUESTION: ...?``
Page 4:
Lemma 3 is new (give a proof) or from a cited paper (and then write [ …])?
Line 42: Move ``\end{lemma}`` before `` To establish our main result, we must first prove the following lemma.``
Page 5:
Line 3 before equation (2): ``for`` instead of ``foll``
Page 6:
Line 59: ``sequences`` instead of ``sequence``
Line 2 from the bottom: ``exists`` should be ``exist``
Page 7:
Line 63: ``With`` instead of ``with``
Second line after (6): replace ``,`` with ``.``
After (7): `` Similarly, Since`` should be `` Similarly, since``
First line after (8): ``exist`` instead of ``exists``
Page 8:
First line after (14): reformulate ``is bounded. From the definition of xn and (4). Let``
Page 9:
Lines 78 and 80: ``conditions`` instead of ``condition``
Page 10:
Line 83: reformulate ``Without loss of generality. Assume``
Page 11:
Line 91: ``condition`` should be ``conditions``
Page 12:
First line: ``From (29) and (30) we get`` or ``It follows from (29) and (30 that`` instead of ``It follows from (29) and (30), we get``
Page 13:
Line 116: ``sequence`` should be ``sequences``
Line 119: ``é`` instead of ``’e``
Line 121: `` Optimality`` should be `` optimality``
Page 14:
In Example 1: reformulate``be the linear spaces elements are`` (``be the linear space whose elements are`` or something else)
Line 2 from the bottom: ``. Define`` should be ``, define``
Page 15
Line 138: ``sequences`` instead of ``sequence``
Line 140: `` lipschitz`` should be ``Lipschitz``
Line 141: ``(i)-(iii)`` instead of ``i)-iii)`` and `` definition`` instead of `` defination``
Lines 145-147: reformulate (now: ``If ... and if ... .``)
First line after (38): ``, this `` should be ``. This``
Line 149: add one space before (
From line 150 to the end: \pi instead of \Pi
In Example 2: ``Let R be the set of all real numbers`` instead of `` Let R be the real number``; ``. Define`` should be ``, define`` (twice times); ``constant`` should be ``constants``; ``as follows:`` instead of ``as follows;``
Page 16
Line 153: ``sequence`` should be ``sequences``
Line 154: ``f_i is satisfies the lipschitz-like`` should be ``f_i satisfies the Lipschitz-like``
Line 155: ``with constants`` instead of ``with constant`` and ``(i)-(iii)`` instead of ``i)-iii)``
Line 157: ``definition`` instead of ``defination``
Line 162: ``References`` is missing
In References section, standardize the items. For example, journal name is complete ([1]) or abbreviated ([2])? In [1] article number and DOI are missing
Comments on the Quality of English LanguageDear authors,
Re-read the paper and correct the English. See, for example, some of the previous comments.
Author Response
Please find the attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsRegarding my first report and the revised version based on it, I think this version is acceptable without any further changes, but the final decision rests with the editor.
Sincerely yours,
Referee
