Abstract Evolution Equations with an Operator Function in the Second Term
Round 1
Reviewer 1 Report
Applications of summation methods for series of root vectors of non-self-adjoint operators to the theory of evolutionary equations in Hilbert spaces are considered. In particular, the author significantly extended the conditions on the operator coefficients of evolution equations and formulated them in terms of operator functions defined on a set of non-self-adjoint operators.
The results of the work are interesting, relevant and will be in demand. The article is written at a high level and deserves publication.
There is a note about the terminology used. The terms "left-hand side" and "right-hand side" do not apply in the context used. Starting with the title of the article, this leads to misunderstanding. The author is encouraged to use generally accepted terminology.
Author Response
Dear referee, I am sincerely grateful to you for your high appreciation of the results and the made remark. Frankly speaking the remark is most valuable for it appeals to the matter that lies on the one hand on the surface but on the other hand it happens that such things can be hardly noticed sometimes! Certainly, I have made the corresponding changes throughout the paper. \\
\noindent Author:\\
Dear referee, I highly appreciate your attention and very grateful to you for the remarks which allow me to see the matter from another point of view and in this way to improve the paper significantly.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Overall, the results appear correct and the proofs are complete. However, some minor remarks are still there.
1) English, especially the usage of 'a/an' and 'the' may be improved.
2) Preliminaries. Does the notion ${\mathbb N}_0$ mean non-negative integers?
3) Lemma 1. It would be better to clarify that $\zeta$ is the ray of the complex plane.
4) Lemma 3. The notion $n_{B^{m+1}}(t)$ is not totally clear for me. It seems, it has never been introduced before.
5) Page 6, line -1. Probably, it is better to say 'an integral'.
6) Page 8, line -4. A comma is necessary after 'As a main result'.
7) Page 8, line -2. 'discus' --> 'discuss'.
8) Page 11, Remark. The reference to a source of the Weiman theorem is redundant here; it is given several lines before.
Author Response
\noindent Author: Dear referee, I am sincerely grateful to you for your attention and the made remarks. I have made necessary changes in accordance with the made remarks. However,
let us consider them consistently.\\
\noindent Referee:
1) English, especially the usage of 'a/an' and 'the' may be improved.\\
\noindent Author: Thank you for a valuable remark, I have checked again the articles and cannot see sharp mistakes, I understand your remark for it is not a secret that there exists duality in using articles. As for a/an, I cannot see misprints. I would be grateful to you if you show them to me.\\
\noindent Referee:\\
2) Preliminaries. Does the notion ${\mathbb N}_0$ mean non-negative integers?\\
\noindent Author: Dear referee, certainly, it does! \\
\noindent Referee:\\
3) Lemma 1. It would be better to clarify that $\zeta$ is the ray of the complex plane.\\
\noindent Author: Thank you for a valuable remark, I have made the corresponding changes.\\
\noindent Referee:\\
4) Lemma 3. The notion $n_{B^{m+1}}(t)$ is not totally clear for me. It seems, it has never been introduced before.\\
\noindent Author: Dear referee, Thank you for a valuable remark, the very notion has been introduced before. You can see it in the preliminary section. \\
\noindent Referee:\\
5) Page 6, line -1. Probably, it is better to say 'an integral'.\\
\noindent Author: Thank you for a valuable remark, I have made the corresponding changes.\\
6) Page 8, line -4. A comma is necessary after 'As a main result'.\\
\noindent Author: Thank you for a valuable remark, I have made the corresponding changes.\\
\noindent Referee:\\
7) Page 8, line -2. 'discus' -- 'discuss'.\\
\noindent Author: Thank you for a valuable remark, I have made the corresponding changes.\\
8) Page 11, Remark. The reference to a source of the Weiman theorem is redundant here; it is given several lines before\\
\noindent Author: Thank you for the remark, I ought to note that the Weiman theorem neither have been previously introduced nor mentioned in the context. Thus, I prefer to preserve it as it is, if you do not mind. \\
\noindent Author: Dear referee, I highly appreciate your attention and very grateful to you for the remarks that allow me to improve the paper significantly!
Author Response File: Author Response.pdf
Reviewer 3 Report
No
Author Response
Dear referee, I am sincerely grateful to your for such high appreciation of the results! It encourages me to work harder and be more peculiar in my writing. Thank you very much, your short comment sounds loudly!\\
\noindent Author: I highly appreciate your attention and enormously grateful to you for the positive review report!
\vspace{0.1 cm}
Sincerely yours Ph.D. Maksim V. Kukushkin
Author Response File: Author Response.pdf
Reviewer 4 Report
Report on axioms-1878138
Manuscript title: "Abstract evolution equations with an operator function at the right-hand side".
Journal: axioms-1878138
There are some minor comments considered as follows and then accepted for publication:
-In introduction section, some new contributions must be cited in the text and inserted in the Refs. List.
- Add reference to equations in this paper.
- Write the physical significance of governing equations.
- Future research direction most be shown in conclusion.
Finally, I recommended the manuscript for publication in axioms-1878138 after releasing the above points.
Author Response
Dear referee, I am sincerely grateful to you for your high appreciation of the results and the made remark. Frankly speaking the remark is most valuable for it appeals to the matter that lies on the one hand on the surface but on the other hand it happens that such things can be hardly noticed sometimes! Certainly, I have made the corresponding changes throughout the paper. \\
\noindent Author:\\
Dear referee, I highly appreciate your attention and very grateful to you for the remarks which allow me to see the matter from another point of view and in this way to improve the paper significantly.
Author Response File: Author Response.pdf
