An Extension on the Local Convergence for the Multi-Step Seventh Order Method with ψ-Continuity Condition in the Banach Spaces

Round 1

Reviewer 1 Report

JOURNAL: Fractal and Fractional

TITLE: An Extension on the Local Convergence for the Multi-Step Seventh Order Method with -Continuity Condition in the Banach Space

AUTHORS: M. T. Darvishi, R. H. AL-Obaidi, Akanksha Saxena, Jai Prakash Jaiswal, and Kamal Raj Pardasani

SUGGESTION: MINOR REVISION

I have reviewed the paper in detail.

The main problems in the convergence of iterative methods are the radius of convergence, the selection of the initial point and the uniqueness of the solution. In this work, the authors focus on these problems via efficient seventh order method by considering the sufficient convergence means, their analysis is applicable to solve such nonlinear problems when both Lipschitz conditions fail without applying higher-order derivatives. Also, they construct a convergence theorem for existence and uniqueness of the solution and provide error bounds. Comparison of convergences of Radius is given by some tables, theoretical results are supported by numerical experiments. This makes the paper more strong.

I only have one remark: is it possible to add some related graphical results (figures) for the given examples. If yes, please add them.

Also, please mention and cite your related paper https://arxiv.org/abs/2112.06177

and explain the differences, if there are.

The organization of the paper is very good. The paper has new and original results.

I recommend this paper to be published.

Comments for author File: Comments.pdf

Author Response

Pointwise Response Sheet on

Comments of Reviewer 1

The pointwise responses on the comments of reviewer 1 on the article entitled “An Extension on the Local Convergence for the Multi-Step Seventh Order Method with ψ-Continuity Condition in the Banach Space by M. T. Darvishi, R. H. AL-Obaidi, Akanksha Saxena, Jai Prakash Jaiswal, and Kamal Raj Pardasani” are as follows:

Comment 1: is it possible to add some related graphical results (figures) for the given examples. If yes, please add them.

Response: Some figures have been added in the numerical examples for showing the significant of the considered discussion over existing one

Comment 2: Also, please mention and cite your related paper https://arxiv.org/abs/2112.06177 and explain the differences, if there are.

Response: The suggested  preprint has been cited.  The presentation of the article specifically abstract, conclusion and numerical representation of the article has been modified in comparison with preprint.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments for author File: Comments.pdf

Author Response

Pointwise Response Sheet on

Comments of Reviewer 2

The point-wise responses on the comments of reviewer 2 on the article entitled “An Extension on the Local Convergence for the Multi-Step Seventh Order Method with ψ-Continuity Condition in the Banach Space by M. T. Darvishi, R. H. AL-Obaidi, Akanksha Saxena, Jai Prakash Jaiswal, and Kamal Raj Pardasani” are as follows:

Comment.1: Check the Equations 6, 7, 8, 9, 22, 23, 24.

Response: The pointed equations have been revisited and typos & unnecessary extra brackets are removed.

Comment 2:-Also should improve language problem.

Response: Language has been improved.

Comment 3:-Conclusion part should be written in detail, i.e., the final results should be explained more detailed form.

Response: The conclusion has been elaborated in some more details.

Author Response File: Author Response.pdf

Reviewer 3 Report

The Authors deal with the scheme for solving non-linear equations given by Xiao and Yin ((4) lines 48-49). In particular, they present ways to find areas of convergence that are superior to other papers.

In my opinion, the paper is interesting and worth publishing in the Fractal Fract journal. However, it needs corrections:

-p.3 lines 60-61. In the equation, under the integral sign - ‘’v(t)’’ should be added

-p.3 lines 71-73. I think that first you should define \psi_0 and then \psi

-p.3 line 72. What is r_0? A comment is needed

-p 7 lines 120-121 eq (43). The sign * next to y should be at the top

-p.8 Example 2, lines 142-143. In the second equation should be y instead of x

-p.9 Example 4. Here x=0 is also a solution of f(x)=0. A comment is needed.

-English language correction is needed

Author Response

Pointwise Response Sheet on

Comments of Reviewer 3

The point-wise responses on the comments of reviewer 3 on the article entitled “An Extension on the Local Convergence for the Multi-Step Seventh Order Method with ψ-Continuity Condition in the Banach Space by M. T. Darvishi, R. H. AL-Obaidi, Akanksha Saxena, Jai Prakash Jaiswal, and Kamal Raj Pardasani” are as follows:

Comment 1: p.3 lines 60-61. In the equation, under the integral sign - ‘’v(t)’’ should be added

Response:The equation has been corrected.

Comment 2:-p.3 lines 71-73. I think that first you should define \psi_0 and then \psi

Response: The notation \psi has been removed from line 71 to avid the confusion.

Comment 3:-p.3 line 72. What is r_0? A comment is needed

Response: This typos error. It is replaced by ρ0.

Comment 4:-p 7 lines 120-121 eq (43). The sign * next to y should be at the top

Response: The suggested typos error has been rectified.

Comment5: -p.8 Example 2, lines 142-143. In the second equation should be y instead of x

Response: The typos error is corrected.

Comment6:-p.9 Example 4. Here x=0 is also a solution of f(x)=0. A comment is needed.

Response: If we take x^*=0, the inverse of first derivative at x^* in example 4 tends to infinite and due to that the equation (19) and (20) will not be fulfilled.

Comment7:-English language correction is needed

Response: The possible modifications have been done,

Author Response File: Author Response.pdf