Cauchy Problem for an Abstract Evolution Equation of Fractional Order
Sergey Piskarev
Round 1
Reviewer 1 Report
The author defined an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, he considered the case when a function has a decomposition in the Laurent series with the infinite principal part and ¯nite regular part. Our central challenge is to improve this result having considered as a regular part an entire function satisfying the special condition of the growth regularity. This manuscript contians publishable results.
Author Response
Dear referee, I highly appreciate your attention and enormously grateful to you for the positive opinion! See the attached file.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
In the paper a method to study Cauchy problems for abstract fractionalevolution equations with the operator function in the second term is invented.
The considered class corresponding to the operator-argument is rather wide and
includes non-selfadjoint unbounded operators that are of a great interest for
applied mathematician. The author, as a main result, represents an approach allowing to
consider a sufficiently wide conditions imposed upon the second term of the
fractional evolution equation. The application part of the paper appeals
to the theory of fractional differential equations and is rather relevant due to this reason. The approach involving an operator-function is novel and rather interesting, the general previously implemented scheme of reasonings contains new technicalities and deserves to be appreciated. As the author correctly mentioned, the made approach allows to obtain a solution for the evolution equation with the operator function in the second term where the operator-argument belongs to a sufficiently wide class of operators. In this regard such well-known operators as the Riesz potential, the Riemann-Liouville fractional differential operator, the Kipriyanov operator, the difference operator and many others can be considered. Moreover, I do think that it is a separate merit for the author being a pure mathematician to have produced an application of the obtained results to engineering sciences connected with electron-induced kinetics of ferroelectrics polarization switching as the self-similar memory physical systems.
Author Response
Dear referee, I highly appreciate your attention and enormously grateful to you for the positive opinion! See the attached file.
Author Response File: Author Response.pdf
Reviewer 3 Report
Please see my report
Comments for author File: Comments.pdf
Author Response
Dear referee, I highly appreciate your attention and enormously grateful to you for the opinion! See the attached file.
Author Response File: Author Response.pdf
Reviewer 4 Report
The connection between the main result of the paper and the Cauchy problem for an abstract evolutionary equation of fractional order should be described more convexly. Namely, what kind of new results can be obtained in this way for equations in a Banach space? One should discuss condition (H1) in more detail and compare the class of these operators with operators generating semigroups.
Author Response
Dear referee, I highly appreciate your attention and enormously grateful to you for the opinion! See the attached file.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
I thanks the author for his excellent revision. The paper can be accepted in this nice form. Just you should use MDPI template and be careful when you use it.
