Monotonicity Arguments for Variational–Hemivariational Inequalities in Hilbert Spaces

Round 1

Reviewer 1 Report

Please see the attached report!

Comments for author File: Comments.pdf

Author Response

I really appreciate the interest of the Referee 1 on my work. I thank him for his careful evaluation of my manucript.

Reviewer 2 Report

*) There are some typos in the text. Please delete them.

*) The authors, in the Introduction of the submitted paper, describe that inequality problems of the form (1) arise in the study of mathematical models which describe the equilibrium of an elastic body in contact with a foundation. If possible, please associate any pertinent references in order to make the work done by the authors attractive also for engineering realities. If possible, proceed equally in similar cases. 

*) In the Introduction, please specify in more detail the contribution of the authors in the submitted paper.

*) If possible, please number all formulas.

*) The proof of Proposition 7 is not very clear. Please re-read it and provide additional details to make it more readable.

*) At the end of the work, it would be appropriate to include a concluding section in which to provide additional comments and details.

*) It is worth noting that variational-hemivariational inequalities governed by both convex and locally Lipschitz functions have a very developed field of applicability in the context of complex fluids (such as, for example, magnetorheological fluids). Furthermore, the use of subdifferentials has allowed to highlight important correspondences between complex (but complete) theoretical models of the "mixture" type with experimental models used in the production of magnetorheological fluids and in the industry that exploits their widespread use. So, for the sake of completeness, I ask the authors to insert a sentence in the text of the paper in order to highlight this peculiarity by putting the following relevant works in the bibliography:

doi: 10.1016/j.ijnonlinmec.2019.103288

ISBN: 978-3-319-44362-1

Author Response

I thank the Referee 2 for his careful reading of my manuscript.

The changes asked by Referee 2 are in red color, so they are easy to be identified. The details are the following :

a) In the Introduction I  provided pertinent references for contact problems in which inequalities of the form (1), (2), (3) are used.
Moreover, I introduced the new references [17] and [19], together with some comments. I specified in more details the novelties of the results in this paper, as well.

b) I provided additional details in the proof of Proposition 7.

c) I introduced a concluding section at the end of the manuscript.

d) I revised the English in the full manuscript.

Reviewer 3 Report

General comments:
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In this paper, the author considers three variational-hemivariational inequalities in a real Hilbert space.

The paper completes some known results regarding the existence, uniqueness and convergence for the mentioned above problems.

The paper is well written and the results are clearly presented.

Specific minor comments:
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"variational-hemivariaional inequality" (the lines 34, 43, 51); 

"variatioanal" (the lines 244, 280, 311); 

"is determied" (line 275);

the sections 4 and 5 (the pages 12 and 14, respectively) have the same title; 

Line 330 and 331: We now use... (two times) - could be reformulated;

Author Response

I thank Referee 3 for his careful evaluation of my manuscript.

The changes asked by Referee 3 are minor. They are in blue color, so it is easy to identify them.