Approximation of the Solution of Split Equality Fixed Point Problem for Family of Multivalued Demicontractive Operators with Application
Round 1
Reviewer 1 Report
In the paper under consideration, the authors have proposed a new viscosity type iterative algorithm is used for obtaining a strong convergence result of split equality fixed point solutions for infinite families of multi-valued demicontractive mappings in real Hilbert spaces and have applied to solve split convex minimization problem.The result obtained is new and useful. The paper is well-written and the proofs are rigorously presented. It is expected that this work will spark the interest for further research in this active domain. I recommend this paper for publication in this journal. After the authors should check all English grammar throughout of this paper.
Author Response
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Reviewer 2 Report
1-Page 2 line 19, multiple full stops.
2-Page 4 line 66, incomplete statement on the sequence of nonnegative real numbers.
3-There is the need to add citations on lemma 1 and lemma 2 or the respective proofs.
4-The correct notation of an empty set should be used throughout the write-up.
5-In page 9, first line, what is lemma 1(a)i?
6-In theorem 1, definitions of omega, A, and B are not defined. Moreover, a clear link to the problem (5) needs to be stated in other to use the accompanying assumptions or conditions therein.
7-In page 11 line 110, what do you mean by " Using lemma 2 in (36)"? (36) is in corollary 1.
Author Response
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Reviewer 3 Report
The comments can be seen in the attached file.
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Reviewer 4 Report
The article presents results that appear in a sense of a movement on a much more general topic: Iterative schemes and convex optimization problems. The authors' approach is based on some classical theories involving projection method on Hilbert space and fixed-point arguments. The authors provide rationale motivations for performing the study based on an interesting, but partial, list of the literature; also, it is of the appropriate length. The authors define terms used in the remainder of the manuscript. The results clearly explained, their order of presentation is parallel to the order of presentation of the methods. Hence, the theoretical results seem to be correct, also because they follow step by step the classical arguments in the literature. The authors have quite appropriately represented the salient points in the articles in the reference list, but there are important references that are not mentioned that should be noted.
In details, my comments on the paper:
1. Quality of writing: good level. There are neither significant typos nor grammar mistakes.
2. Quality of content: correct with sufficient level of novelty. The main result (i.e., Theorem 1) is well developed and well presented. At the best of my knowledge, the proof is correct and the result is new.
3. Application: relevant. The authors solve split convex minimization problem, suitable in various fields of research (see Theorem 2).
I support the acceptance of the paper. However, I suggest the authors to cite and discuss properly in the introduction and/or in the application section the following recent and strongly related papers which deal with the extrapolation and extra-gradient methods, and provide a numerical analysis of the proposed algorithms:
Demonstratio Mathematica 2022; 55: 193–216
Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces
G.N. Ogwo, T.O. Alakoya, O.T. Mewomo
In this paper, the authors discuss an inertial iterative algorithm with self-adaptive step size for approximating a common solution of finite family of split monotone variational inclusion problems and fixed point problem of a nonexpansive mapping between a Banach space and a Hilbert space.
Journal of Convex Analysis 2018; 25:701-715
Fixed point iterative schemes for variational inequality problems
E. Toscano, C. Vetro
In this work the authors apply fixed point iterative schemes to variational inequality problems, via admissible perturbations of projection operators in real Hilbert spaces. They discuss convergence properties. In particular, the authors involve a class of α-co-coercive operators with application to general equilibrium problems.
Demonstratio Mathematica 2022; 55: 297–314
On solving pseudomonotone equilibrium problems via two new extragradient-type methods under convex constraints
C. Khunpanuk, N. Pakkaranang, N. Pholasa
In this study the authors discuss certain proximal-type algorithms for solving equilibrium problems in real Hilbert space. The algorithms are analogous to the well-known two-step extragradient algorithm for solving the variational inequality problem in Hilbert spaces. The proposed iterative algorithms use a step size rule based on local bifunction information instead of the line search technique. Several computational experiments are depicted to demonstrate the efficiency and effectiveness of the proposed algorithms.
Rend. Circ. Mat. Palermo, II. Ser 71, 325–348 (2022). https://doi.org/10.1007/s12215-021-00608-8
An Iterative method for split equality variational inequality problems for non-Lipschitz pseudomonotone mappings
K.M.T. Kwelegano, H., Zegeye, O.A. Boikanyo
The authors discuss an algorithm for approximating solutions of split equality variational inequality problems. A convergence theorem of the proposed algorithm is established in Hilbert spaces under the assumption that the associated mapping is uniformly continuous, pseudomonotone and sequentially weakly continuous. The behavior of the convergence of the algorithm is discussed in the numerical part.
Finally, I suggest a careful reading to correct some minor typos. For example, I noted:
pg2, line -18: correct "there exists";
pg2, line -3: correct "mapping";
pg3, line +3: correct "follows.";
pg5, line+2: correct the itemization;
and so on.
Author Response
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Author Response File: Author Response.docx
Round 2
Reviewer 3 Report
The detailed comments can be seen in the attached file.
Comments for author File: Comments.pdf
Author Response
Please see attachment.
Author Response File: Author Response.docx
Round 3
Reviewer 3 Report
This paper has some serious flaws. Therefore, the paper should not be accepted in this journal. The detailed comments can be seen in the attached file.
Comments for author File: Comments.pdf
Author Response
see cover letter
Author Response File: Author Response.docx
