Accurate Estimations of Any Eigenpairs of N-th Order Linear Boundary Value Problems

Round 1

Reviewer 1 Report

The authors present an interesting numerically-analytical approach to one-dimensional eigenvalue problem with a sign-regular kernel. This paper develops the series of more earlier works by Stepanov, Levin and etc. On my opinion the derivations and proofs are seem correct, and the study is quite original and novel. The reported results should be of interest to the readers. I recommend the publication of the manuscript.

Author Response

Dear Referee.

Many thanks for dedicating your team to review our paper.

Happy to see you enjoyed it.

Cheers,

Pedro Almenar and Lucas Jódar

Reviewer 2 Report

In this paper some eigenvalues and eigenfunctions of $n-$boundary value problems with singular kernel. I think that the paper is well-organized and the significance of the main ideas are interesting. The arrangement of the material is suitable. The originality and technical quality are good. In my opinion, the manuscript can be published in present at “Fractal and Fractional”.

Author Response

Dear Referee.

Many thanks for dedicating your time to review our paper.

Happy to see you enjoyed it.

Cheers,

Pedro Almenar and Lucas Jódar

Reviewer 3 Report

    Dear Editor
    The article is presented in logical way and overall written well but the following remarks can be made on the article:

1.Line 1, (calculate) can be replaced with (calculates).
2.Line 19, (Throughout the paper) can be replaced with (Throughout the paper,).
3.Line (34), (for very p) can be replaced with (for every p).
4.Line (72), (associated to) can be replaced with (associated with).
5.Line (84), ( into int(p) ) can be replaced with ( into an int(p) ).
6.Line (107), (In the Section 2) can be replaced with (In Section 2).
7.Line (108), (the subsection 2.1) can be replaced with (subsection 2.1).
8.Line (109), (the subsection 2.2) can be replaced with (subsection 2.2).
9.Line (110), (to each cone,) can be replaced with (to each cone).
10.Line (111), (The section 3) can be replaced with (section 3).
11.Line (112), (Finally the Section 4) can be replaced with (Finally, Section 4).
12.Line (147), (For the function y we define the one-sided limit) can be replaced with (For the function, y we define the one-sided limit).

    
    Additional Comments

1.1)   On row 7-Page 1: "order linear differential equation, two-point boundary value problem, sign-regular kernel, eigenvalue" is unrelated to the paper subject. 
2.2)   On rows 76-Page 4: Replace " ..then" with " ..,then".
3.3)   Before row 144-Page 7: Replace " Ly = Lny " with " and Ly = Lny,".
4.4)   After row 167-Page 6 in Theorem 2: Replace " itself then " with " itself, then".
5.5)   After row 198-Page 11: Replace " From (1)-(2) one" with " From (1)-(2), one".
6.6)   Before row 204-Page 12: Replace " from [1, Equation (12.12)] we know" with " from [1, Equation (12.12)], we know".
7.7)   On row 226-Page 13: What do you mean by " r(Mp,Mjpu) and r(Mp,Mjpu)" ?.
8.8)   On Row 235-Page 14: Replace "..then its Collatz-Wielandt" with "…,then its Collatz-Wielandt".
9.9)   Page 16 the row under inequality 73: Replace "… then the determinant" with "…,then the determinant".
10.10)   The format of references should be standardized, for example reference (3,9,11)

    The article can be recommended for publication after minor revisions.

Author Response

Dear Reviewer.

Thanks a lot for the time dedicated to review our paper and for the corrections you have proposed.

Let us go through them one by one:

1. We have rephrased the statement to 'a method to bound and to calculate any eigenvalues...', instead of using the proposal 'calculates', as we are talking of the purpose of the method.
2. Corrected as per your proposal.
3. Corrected as per your proposal.
4. Corrected as per your proposal.
5.  M^q maps P\{0} into int(P), which is a subset of the same cone P. We are not sure about your point on replacing by 'into an int(P)'.
6. Corrected as per your proposal.
7. Corrected as per your proposal.
8. Corrected as per your proposal.
9. Corrected as per your proposal.
10. Corrected as per your proposal.
11. Corrected as per your proposal.
12. Corrected as per your proposal.

As for your additional comments:

1.1) We have replaced commas by semicolons in the list of keywords. We are not sure if you meant anything else, since the the purpose of the paper is to calculate eigenvalues of two-point boundary value problems whose differential equations are linear and of n-th order, and whose kernels are sign-regular.

2.2) Corrected as per your proposal.

3.3) Corrected as per your proposal.

4.4)  Corrected as per your proposal.

5.5)  Corrected as per your proposal.

6.6)  Corrected as per your proposal.

7.7) The second r(M_p,M^j_pu) should be the upper Collatz-Wielandt number instead of (again) the lower one. It is a typo in the text corrected in the new version. 

8.8)  Corrected as per your proposal.

9.9)  Corrected as per your proposal.

10.10) Amended. It was due to an wrong 'and' in the authors part of some bibliography entries.

Please let us know if you find anything missing.

Best regards,

Pedro Almenar and Lucas Jódar

Reviewer 4 Report

This paper proposes a method to bound and compute any pair of eigenvalues and eigenfunctions of n-th order boundary value problems with very general two-point boundary conditions, extending previous works of the authors where only the principal eigenpair was considered. The capabilities and limitations of the proposed method are critically discussed, and possible applications and extensions are suggested.   

The paper is very well written and the mathematics seem to be correct. I find that this is a very good paper, and I recommend it to be published.

Some minor comments and suggestions are indicated below.

Lines 14-15. [12, page 47] could also be referred to for definition of sign-regular kernels.

Line 36. “eigenfunctions functions”.

Check for superfluous commas at the end of some expressions (e.g., (27), (30) in page 8).

Check for possibly superfluous “the” (e.g., some titles or nouns with numbers).

Line 282. A reference for the result in Theorem 6 could be included.

Line 387. Include Issue (1), as page numbers are within-issue.

Line 407. Should be: Total Positivity, Interpolation by Splines, and Green’s Functions of Differential Operators.

Author Response

Dear Reviewer.

Thanks for the time dedicated to review our paper and for proposing some improvements. Happy to see you enjoyed it.

As for your comments:

• Lines 14-15. [12, page 47] could also be referred to for definition of sign-regular kernels - > Reference added in the text.
• Line 36. “eigenfunctions functions”. -> Superfluous "Functions" removed from the text.
• Check for superfluous commas at the end of some expressions (e.g., (27), (30) in page 8). -> Checked and removed (equation (53) was also affected).
• Line 282. A reference for the result in Theorem 6 could be included. -> A reference to [13, Theorem 4] and [1, Theorem 8] has been included.
• Line 387. Include Issue (1), as page numbers are within-issue. -> The original .bib file included the issue but the BibTex package does not display it, so the Issue 1 has been included in the reference manually (not ideal, but...).
• Line 407. Should be: Total Positivity, Interpolation by Splines, and Green’s Functions of Differential Operators. -> Paper title amended accordingly.

Best regards,

Pedro Almenar and Lucas Jódar

Round 2

Reviewer 3 Report

I recommend publishing this paper in Mathematics