Fixed Point Results on Multi-Valued Generalized (α,β)-Nonexpansive Mappings in Banach Spaces

Round 1

Reviewer 1 Report

I would like to see examples of multivalued generalized (a, b)-nonexpansive operators more typical for applications. Acting in both finite-dimensional and infinite-dimensional spaces.

Author Response

Firstly, we shall express our sincere thanks to this reviewer for carefully observing our paper during the review process. In the first version of the paper, we have already provided an example of multi-valued generalized (\alpha, \beta)-nonexpansive mappings. We have proved that this example exceeds the corresponding classes of  some nonexpansive mappings of the existing literature. We have also showed that the proposed scheme of this example is more effective than the many others schemes.

Now in the revised version of the paper, we have constructed another example, which illustrates the wideness of the class of (\alpha, \beta)-nonexpansive mappings.

Thank you for the review.

Reviewer 2 Report

The article presents results that appear in a sense of a movement on a much more general topic: Fixed point theorems and fixed-point iterative methods. The authors' results are developed on some classical  approaches involving fixed-point techniques together with the theory of operators in Banach spaces. Also the authors provide a comparative study with other similar algorithms in the literature. The theoretical results seem to be correct, even if they follow step by step the classical arguments in previous papers. Moreover, they are of a certain interest for researchers working on related topics, as variational inequalities and equilibrium problems approached via the fixed point theory.

In details, my comments on the paper:

1. Quality of writing: good level. There are neither significant typos nor grammar mistakes.
2. Main results: correct but average level novelty. That is, results are quite standard but well developed and well presented. At the best of my knowledge, the proofs are correct and the results are new.
3. Case study: standard. The authors propose an illustrative example, and discuss the convergence of the algorithm in comparison with the similar algorithms in the literature. 
4. Literature: adequate. There is really a vast literature on the topic, so it is difficult to cover everything. By the way, I would like ask the authors  to cite and discuss properly in the introduction the results contained in the following two very recent papers:

[R1] Nopparat Wairojjana, Nuttapol Pakkaranang, and Nattawut Pholasa
Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces
Demonstratio Mathematica 54: 110–128 (2021)

[R2] Sabiya Khatoon and Izhar Uddin 
Convergence analysis of modified Abbas iteration process for two G-nonexpansive mappings
Rendiconti del Circolo Matematico di Palermo Series 2; 70: 31–44 (2021)

Indeed, [R1,R2] pose some interesting problems in dealing with iterative fixed-point schema, respectively in Hilbert spaces and in uniformly convex Banach spaces. [R1] concerns variational inequalities. 

Author Response

We want to express our sincere thank to the reviewer for the constructive review report. We have done the following improvements in the revised version.

• We have checked the paper carefully and corrected and improved any typo found in the initial version.
• We have again checked the proofs of the main results and found them corrected. However, where some little change needed, then it has been revised.
• We have inserted an illustrate example in the revised version which will helps the reader to know about the wideness of the proposed class of mappings.
• We have checked some other related paper and also the suggested ones, and we cited those who are more related.

We hope that the revised version will be ready for the publication. Thank you for the review.

Reviewer 3 Report

The authors can find several comments and suggestions for improvement in the accompanying report.

Comments for author File: Comments.pdf

Author Response

The authors of the manuscript are very much thankful for the beautiful and detailed review report. We have executed all the suggestions and modifications.

We hope that the revised version will be of the acceptable level. Thank you once again.

Round 2

Reviewer 3 Report

The authors can find several comments in the accompanying referee report. A number of corrections and changes are still required.

Comments for author File: Comments.pdf

Author Response

All the suggestions are executed successfully, for detail please check the updated version of the paper. I am thankful to the reviewers for their detail comments. The paper has been improved very much due to their comments.