Modified Inertial-Type Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Finite Bregman Relatively Nonexpansive and Demicontractive Mappings

Round 1

Reviewer 1 Report

1.     The original contributions need to be much better presented in the last paragraph of section \INTRODUCTION". All improvements, if they are, and new results must be described in this paragraph. The advantages of the work are not discussed in the text. Indeed, introduction needs to enrich the readers with state-of-the-art works, which gives the reader a clear vision of the gap that those studies have not addressed and covered in this study.

 

2.     The Abstract should modified by adding advantages of the proposed method.

 

3.     Check the manuscript carefully for typos and grammatical errors.

 

4.     Please clarify the novelty of this paper with respect to the published paper.

 

5.      The references list is not at all updated with latest developments and publications. I suggest   the authors to keep up to date with the relevant literature such as:

 

“The analysis of fractional-order nonlinear systems of third order KdV and Burgers equations via a novel transform”  “Evaluation of time-fractional Fisher’s equations with the help of analytical methods”

 

6. Please report the CPU running times in given numerical examples.

7. Section Conclusion should be elaborated more in detail.

8. Please improve the quality of given Figures.

 

9. At the beginning of the numerical results section, authors should present the configuration of the personal computer used to perform the simulation results.

Author Response

Referee 1 comments and our responses:

1. The original contributions need to be much better presented in the last paragraph of section “INTRODUCTION”. All improvements, if they are, and new results must be described in this paragraph. The advantages of the work are not discussed in the text. Indeed, introduction needs to enrich the readers with state-of-the-art works, which gives the reader a clear vision of the gap that those studies have not addressed and covered in this study.

   Our response:  As mentioned in the final paragraph 2 of section “INTRODUCTION” (see Page 5 of the current version), the existing method in [31] is most closely related to our proposed method, that is, the hybrid projection method for resolving a single VIP with FPP constraint in [31] is extended to develop our modified inertial-type subgradient extragradient method for resolving a pair of VIPs with CFPP constraint. Compared with the corresponding results in [31], our results improve, extend and develop them in the two aspects below: (i) the problem of finding a solution of a single VIP with FPP constraint (involving a Bregman relatively nonexpansive mapping) in [31] is extended to develop our problem of finding a solution of a pair of VIPs with CFPP constraint (involving finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping); (ii) the hybrid projection method with line-search process in [31] is extended to develop our modified inertial-type subgradient extragradient method with line-search process.

As mentioned in the final paragraph 1 of section “INTRODUCTION” (see Page 5 of the current version), we have provided numerical example to show the competitive advantage of our proposed algorithms over the existing algorithms, e.g., the ones in [31]. In fact, we have provided the illustrative example of a pair of VIPs with CFPP constraint in Section 4. Note that the existing method in [31] is only utilized to solving a single VIP with FPP constraint. Hence there is no way for this method to handle the numerical example in Section 4, that is, it is un-valid for a pair of VIPs with CFPP constraint. But our suggested method can settle the illustrative example in Section 4. This ensures the competitive advantage of our proposed algorithms over the existing algorithms.

2. The Abstract should be modified by adding advantages of the proposed method.

   Our response:  Yes. We have added and inserted the following sentences in the end of the Abstract:

   “This paper reveals the competitive advantage of our proposed algorithms over the existing algorithms, that is, the existing hybrid projection method for a single VIP with FPP constraint is extended to develop our modified inertial-type subgradient extragradient method for a pair of VIPs with CFPP constraint.”

3. Check the manuscript carefully for typos and grammatical errors.

   Our response:  Yes. We have gone through the whole paper and corrected them.

4. Please clarify the novelty of this paper with respect to the published paper.

   Our response:  Yes. As mentioned in the Remark 3.1 (see Page 26 of the current version), it can be easily seen that the existing method in [31] is most closely related to our proposed method, that is, the hybrid projection method for resolving a single VIP with FPP constraint in [31] is extended to develop our modified inertial-type subgradient extragradient method for resolving a pair of VIPs with CFPP constraint. Compared with the corresponding results in [31], our results exhibit the novelty below:

  First, the problem of finding a solution of a single VIP with FPP constraint (involving a Bregman relatively nonexpansive mapping) in [31] is extended to develop our problem of finding a solution of a pair of VIPs with CFPP constraint (involving finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping).

 Second, the hybrid projection method with line-search process in [31] is extended to develop our modified inertial-type subgradient extragradient method with line-search process.

   As mentioned in the final paragraph of Section 4 (see Page 29 of the current version), we have provided numerical example to show the competitive advantage of our proposed algorithms over the existing algorithms, e.g., the ones in [31]. In fact, we have presented the illustrative example of a pair of VIPs with CFPP constraint in Section 4. Note that the existing method in [31] is only utilized to solving a single VIP with FPP constraint. Hence there is no way for this method to handle the numerical example in Section 4, that is, it is un-valid for a pair of VIPs with CFPP constraint. But our proposed method can settle the illustrative example in Section 4. This reveals the competitive advantage of our proposed algorithms over the existing algorithms.

5. The references list is not at all updated with latest developments and publications. I suggest the authors to keep up to date with the relevant literature such as:

“The analysis of fractional-order nonlinear systems of third order KdV and Burgers equations via a novel transform” and “Evaluation of time-fractional Fisher’s equations with the help of analytical methods”

   Our response:  Yes. As mentioned in paragraph 1 of Section 1 (see Page 3 of the current version), during the past few decades, the fixed point theory has played a vital part in solving many problems arising in nonlinear analysis and optimization theory such as differential hemivariational inequalities systems (see [35]), monotone bilevel equilibrium problems (see [36]), generalized global fractional-order composite dynamical systems (see [37]), generalized time-dependent hemivariational inequalities systems (see [38]), optimal control of feedback control systems (see [39]) and so on. Please the authors add and insert such information in this article.

[35]L.C. Ceng, C.F. Wen, Y.C. Liou, J.C. Yao, A general class of differential hemivariational inequalities systems in reflexive Banach spaces, Mathematics 2021, 9 (24), 3173; DOI: 10.3390/math9243173

[36]L. He, Y.L. Cui, L.C. Ceng et al., Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule, J. Inequal. Appl. 2021, Paper No. 146, 37 pp.

[37]L.C. Ceng, N.J. Huang, C.F. Wen, On generalized global fractional-order composite dynamical systems with set-valued perturbations, J. Nonlinear Var. Anal. 2022 (6) 149-163.

[38]L.C. Ceng, Y.X. Fu, J. Yin et al., The solvability of generalized systems of time-dependent hemivariational inequalities enjoying symmetric structure in reflexive Banach spaces, Symmetry 2021, 13 (10), 1801; DOI: 10.3390/sym13101801

[39]L.C. Ceng, Z. Liu, J.C. Yao, Y. Yao, Optimal control of feedback control systems governed by systems of evolution hemivariational inequalities, Filomat 32 (2018) 5205-5220.

6. Please report the CPU running times in given numerical examples.

   Our response:  Without doubt, it is very important that the CPU running time in given numerical examples is reported. But the investigation on the iterative algorithms and convergence criteria for a pair of VIPs with CFPP constraint in p-uniformly convex and uniformly smooth Banach space has just begun at present. Thus, the CPU running time in given numerical examples needs to be further considered later on.

7. Section Conclusion should be elaborated more in detail.

   Our response:  To elaborate Section Conclusion more in detail, we have added and inserted the following paragraph in Section 5 (see Page 30 of the current version):

“Compared with the corresponding results in [31], our results reveals the novelty as follows: (a) the problem of finding a solution of a single VIP with FPP constraint (involving a Bregman relatively nonexpansive mapping) in [31] is extended to develop our problem of finding a solution of a pair of VIPs with CFPP constraint (involving finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping); (b) the hybrid projection method with line-search process in [31] is extended to develop our modified inertial-type subgradient extragradient method with line-search process.”

8. Please improve the quality of given Figures.

   Our response:  Okay. Many thanks.

9. At the beginning of the numerical results section, authors should present the configuration of the personal computer used to perform the simulation results.

   Our response:  Without question, it is quite important that the configuration of the personal computer used to perform the simulation results is presented. But the investigation on the iterative algorithms and convergence criteria for a pair of VIPs with CFPP constraint in p-uniformly convex and uniformly smooth Banach space has just begun at present. Thus, the configuration of the personal computer used to perform the simulation results needs to be further explained later on.

Author Response File: Author Response.pdf

Reviewer 2 Report

The article is well-presented, and the results are proved in a traditional way by defining sequences using different algorithms and showing their convergence. Overall, the work is mathematically sound. 

Author Response

   Referee 2 comments and our responses:  The article is well-presented, and the results are proved in a traditional way by defining sequences using different algorithms and showing their convergence. Overall, the work is mathematically sound. 

   Our response:  Okay. Many thanks.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

I rejected this paper.
My comments is not clear write one by one and send highlight files.

remove this references 

[35]L.C. Ceng, C.F. Wen, Y.C. Liou, J.C. Yao, A general class of differential hemivariational inequalities systems in reflexive Banach spaces, Mathematics 2021, 9 (24), 3173; DOI: 10.3390/math9243173

[36]L. He, Y.L. Cui, L.C. Ceng et al., Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule, J. Inequal. Appl. 2021, Paper No. 146, 37 pp.

[37]L.C. Ceng, N.J. Huang, C.F. Wen, On generalized global fractional-order composite dynamical systems with set-valued perturbations, J. Nonlinear Var. Anal. 2022 (6) 149-163.

[38]L.C. Ceng, Y.X. Fu, J. Yin et al., The solvability of generalized systems of time-dependent hemivariational inequalities enjoying symmetric structure in reflexive Banach spaces, Symmetry 2021, 13 (10), 1801; DOI: 10.3390/sym13101801

[39]L.C. Ceng, Z. Liu, J.C. Yao, Y. Yao, Optimal control of feedback control systems governed by systems of evolution hemivariational inequalities, Filomat 32 (2018) 5205-5220.

 Moderate editing of English language required

Author Response

Reviewer 1 comments (Round2) and our response.

• Comments:  My comments: not clear to write one by one and please send highlight files.

Please remove these references

[35]L.C. Ceng, C.F. Wen, Y.C. Liou, J.C. Yao, A general class of differential hemivariational inequalities systems in reflexive Banach spaces, Mathematics 2021, 9 (24), 3173; DOI: 10.3390/math9243173

[36]L. He, Y.L. Cui, L.C. Ceng et al., Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule, J. Inequal. Appl. 2021, Paper No. 146, 37 pp.

[37]L.C. Ceng, N.J. Huang, C.F. Wen, On generalized global fractional-order composite dynamical systems with set-valued perturbations, J. Nonlinear Var. Anal. 2022 (6) 149-163.

[38]L.C. Ceng, Y.X. Fu, J. Yin et al., The solvability of generalized systems of time-dependent hemivariational inequalities enjoying symmetric structure in reflexive Banach spaces, Symmetry 2021, 13 (10), 1801; DOI: 10.3390/sym13101801

[39]L.C. Ceng, Z. Liu, J.C. Yao, Y. Yao, Optimal control of feedback control systems governed by systems of evolution hemivariational inequalities, Filomat 32 (2018) 5205-5220.

2) Response:  Okay. We have provided highlight files and removed the references as above; please read the attached current version.

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

Accept