The Sign of the Green Function of an n-th Order Linear Boundary Value Problem
Round 1
Reviewer 1 Report
The present paper deals with the results on the sign of G(x, t), the Green function associated to the problem (4). As authors mention in the section Discussion, the paper extends the results of Eloe and Ridenhour published in 1994. The state of the art is not very well presented based on the references where we can find only 3 papers that are published in last 5 years. I strongly recommend to authors to reconsider the introduction in order to update the state of the art and to enlarge the list of references.
The results are clearly presented and authors provide a strong mathematical content. I recommend the publication of this paper after minor revision
Author Response
Dear Reviewer;
many thanks for your comments.
Based on them we have incorporated the following changes to the paper:
- We have added references [18]-[21] published in the last five years and which provide some information about the usage of the sign of the Green function in non-linear problems. This has also been discussed in the lines 41-45.
- We have added references [23]-[26] (one of them published in 2010 and the other three in the last three years) about fractional boundary value problems in which the knowledge of the sign of the Green function is used at some points of the proofs. This topic is commented in lines 276-282.
Let us highlight that, in general, the vast majority of the recent papers do not provide insights about the sign of the solutions of generic boundary value problems but they focus on the calculation of the Green functions and, based on that, determine their properties. This is especially clear in the case of the fractional boundary value problems, where the resolution of the fractional equation is only possible in the simplest cases. Although these approaches are understandable we believe that a more general like the one presented in this paper is far more powerful.
In addition, as a result of other Reviewers' comments:
- In lines 34-41 we have provided more details on the technical reasons why knowing the sign of the partial derivatives of the Green functions is relevant for the problem (6).
- In lines 66-70 we have provided some examples of (α,β) and calculated the associated variables αA, βB, K(α,β), S(α) and S(β), in order to provide a better understanding of the latter.
- In lines 265-267 we have emphasized the importance of knowing the sign of as many partial derivatives of the Green functions as possible so as to allow higher values of μ in the problem (6).
- In lines 274-282 we have described possible lines of extension of this approach.
Please let us know if all these changes cover your expectations.
Best regards,
Pedro Almenar & Lucas Jódar
Reviewer 2 Report
The paper under review deals with the sign of the Green function and its partial derivatives of a boundary value problem with two-point boundary conditions related to an n-th order ordinary differential equation.
The paper mainly refine the past result contained in reference [1], specifying some proofs of [1], and adding informations on the derivative sign of the Green's function which was missing in paper [1].
The paper is well written and as far as I checked it looks correct. Maybe some more information on the reason why the Authors need the sign information of the Green's function derivative would help the reader to better understand the novelty of the paper.
I have also one question about line 37 at pag. 3. It is not clear to me the definition of K(\alpha, \beta) and I would like to better understand. Maybe some extra word would help.
Said this, after the above mentioned improvements, the paper can be accepted for publication.
Author Response
Dear Reviewer;
many thanks for your comments.
Based on them we have incorporated the following changes to the paper:
- In the lines 34-41 of the Introduction we have provided more details on the technical reasons why knowing the sign of the partial derivatives of the Green functions is relevant for the problem (6).
- In the lines 41-45 of the Introduction we have discussed the applicability of the sign of the Green functions and its derivatives to the non linear version of (6), namely the new equation (9), which was studied in the new references [18]-[21], all of them published in the last five years.
- In the lines 66-70 of the Introduction we have provided some examples of (α,β) and calculated the associated variables αA, βB, K(α,β), S(α) and S(β), in order to provide a better understanding of the latter.
- In the lines 265-267 of the Discussion we have emphasized the importance of knowing the sign of as many partial derivatives of the Green functions as possible so as to allow higher values of μ in the problem (6).
In addition, as a result of other Reviewers' comments:
- In the lines 274-282 of the Discussion, we have described possible lines of extension of this approach.
- We have added references [23]-[26] (one of them published in 2010 and the other three in the last three years) about fractional boundary value problems in which the knowledge of the sign of the Green function is used at some points of the proofs. This topic is commented in lines 276-282, although a remark is done in that in all these cases an explicit solution of the fractional boundary value problem needs to be calculated, as opposed to our paper where the signs are provided without solving any differential equation. In general this is a shortcoming of most of the recent work in Green functions for fractional boundary value problems.
Please let us know if these changes cover your expectations.
Best regards,
Pedro Almenar & Lucas Jódar
Reviewer 3 Report
In this paper the Authors propose results about sign of the Green function of an n-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The results provide information about the sign and dependence on the extremes a and b of the Green function of the problem and its derivatives when the two-point boundary conditions are admissible, property which encompasses many types of boundary conditions usually covered in the literature.
I found that this paper is very interesting and that the obtained results are very promising, however in order to further improve I would only recommend to improve the conclusions and more references on the background.
Author Response
Dear Reviewer;
many thanks for your comments.
Based on them we have incorporated the following changes to the paper:
- In the lines 34-41 of the Introduction we have provided more details on the technical reasons why knowing the sign of the partial derivatives of the Green functions is relevant for the problem (6).
- In the lines 265-267 of the Discussion we have emphasized the importance of knowing the sign of as many partial derivatives of the Green functions as possible so as to allow higher values of μ in the problem (6).
- In the lines 274-282 of the Discussion, we have described possible lines of extension of the approach covered in this paper.
- We have added references [18]-[21], all of them published in the last five years, which deal with the non linear version of (6), namely the new equation (9). We have discussed the applicability of the information about sign of the Green functions and its derivatives to this non linear problem in the lines 41-45 of the Introduction.
- We have added references [23]-[26] (one of them published in 2010 and the other three in the last three years) about fractional boundary value problems in which the knowledge of the sign of the Green function is used at some points of the proofs. This topic is commented in lines 276-282 of the Discussion, although a remark is done in that in all these cases an explicit solution of the fractional boundary value problem needs to be calculated, as opposed to our paper where the signs are provided without solving any differential equation. In general this is a shortcoming of most of the recent work in Green functions for fractional boundary value problems.
In addition, as a result of other Reviewers' comments:
- In the lines 66-70 of the Introduction we have provided some examples of (α,β) and calculated the associated variables αA, βB, K(α,β), S(α) and S(β), in order to provide a better understanding of the latter.
Please let us know if these changes cover your expectations.
Best regards,
Pedro Almenar & Lucas Jódar
