On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space

Round 1

Reviewer 1 Report

The existence and uniqueness problems are studied for some types of fractional integral Riemann - Liouville operators with the kernel satisfying Sonin type conditions. The results are applied for some specific types of kernels.

The obtained results (Theorem 1 and its corollaries) appear novel and correct. The manuscript can be accepted with minor corrections.

However, some minor remarks are still there (I address to the author directly):

1) Page 3, line 6. The equality looks a bit weird. Do you just mean  that all a_i with negative indices i are zeros?

2) Page 4, line -9. Do you men that your conditions imply that all integrals converge?

3) Page 6, line 2. What do you mean by "if their meaning is quite clear"? 

4) Page 10, line -9. I guess there must be \eta' instead of \eta under the integral.

5) Page 11, line 9. "a reader" --> "readers' "

6) Page 11, Corollary 1 "Conditions of Theorem 1 made except of (ii)". Do you mean just the condition (i)?

7) Page 11, line -3 A quasi-solution of what?

8) Page 13, line -9 "Matrix" --> "matrix"

9) Page 14, line -5 "wether" --> "whether"

10) Page 22, line 14 "letter" --> "latter"

Author Response

 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.  Moreover, I have significantly changed   subsection 3.4 and  added subsection 3.5.
 Let us consider the made remarks  consistently.

Author Response File: Author Response.pdf

Reviewer 2 Report

The results are very interesting and the proofs are correct. My opinion is that this article must be accepted for publication in Fractal and Fractional.

The author is invited to take into account some minor changes.

Comments for author File: Comments.pdf

Author Response

 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.  Moreover, I have significantly changed  subsection 3.4 and  added subsection 3.5.
 Let us consider the made remarks  consistently.

Author Response File: Author Response.pdf