Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative
Round 1
Reviewer 1 Report
The paper continues a series of works on proving the existence of solutions of fractional differential equations and inclusions with anti-periodical boundary conditions moving forward from Riesz-Caputo and psi-Caputo derivatives to the fractional inclusions with newly introduced psi-Riesz-Caputo derivative.
The main novelty of the paper is a presentation and usage of a new definition of fractional derivative, namely the psi-Riesz-Caputo derivative.
The conditions of the existence of solutions are obtained using fixed point theorems within the general approach used previously for other similar problems, particularly, in the works cited in the reviewed paper.
The paper is well written and presents new valuable results. However, Introduction structure can be improved rearranging paragraphs to make it more logical. At least, from my point of view, a paragraph with general considerations about fractional calculus (starting from the line 31) will look more appropriate at the beginning of the Introduction.
Author Response
We rearranged our introduction as your suggestions in order to let it seems more logic. Please see the attachedment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Please expand the abstract section. In line 12 you wrote: "By means of some standard fixed point theorems..." What theorems did you mean?
In the introduction section, provide more background about the problem you are considering.
Please expand the application section. Add more examples and compare with others.
Please add the conclusion section.
Also, I found the following errors:
Formulas in definition 1 are Caputo fractional derivate but not integral;
In definition 3 there is an error in the word «fractional».
Author Response
(1) We add the tools of this paper in abstract.
(2) We add line 49-55 as the background about the problem in the introduction.
(3) We expand our applications section.
(4) We add the conclusion section.
(5) We make it correct in Definition I （ derivative）and Definition 3 ( fractional)
Author Response File: Author Response.pdf
Reviewer 3 Report
Referee's Report:
Line 37 ``papers"
Line 39 ``Liouville"
Line 43 ``newly"
Line 45 ``equation"
Line 51 ``involving the"
Line 68 ``Section 3"
Line 70 ``of our problem"
Line 77 ``notation" ( this word in English is both singular and plural)
Line 78 put in a space before ``and"
Line 109 ``If we take"
Lines 114-116 in four places put a space after ``Y"
Line 118 ``in the remainder of this paper"
Line 131 `type"
Line 139 ``We say N"
Line 142 put in a space before last ``t"
Line 168 put in a space before ``g"
Lines 196-197 fix this up
Line 199 ``defined by"
Line 203 ``By a chain rule"
Line 209``Lemma 6"
Lines 213-214 fix this up
Line 234 this line runs into the margin
Line 264 ``According to the"
Line 266 ``we"
Line 291 fix this up
Lines 332-333 use more sentences here
Line 345 this line rums into the margin
Line 352 replace ``by" by ``since"
Lines 364-365 fix this up
Line 367 ``We omit the remainder of the proof"
Line 385 this line rums into the margin
Line 386 ``the fractional"
Throughout the references you should put a space for example in line 404 you should have a space before ``R."
Author Response
Line 37 ``papers", see new line 24
Line 39 ``Liouville", see new line 26
Line 43 ``newly", see new line 30
Line 45 ``equation", see new line 32
Line 51 ``involving the", see new line 39
Line 68 ``Section 3", see new line 73
Line 70 ``of our problem", see new line 75
Line 77 ``notation", see new line 82
Line 78 put in a space before ``and" , see new line 83
Line 109 ``If we take", new line 116
Lines 114-116 in four places put a space after ``Y", see new lines120-124
Line 118 ``in the remainder of this paper", see new line 126
Line 131 `type", see new line 139
Line 139 ``We say N", see new line 147
Line 142 put in a space before last ``t", see new line 150
Line 168 put in a space before ``g", see new line 176
Lines 196-197 fix this up, see new line 204
Line 199 ``defined by", see new line 206
Line 203 ``By a chain rule", see new line 210
Line 209``Lemma 6", see new line 216
Lines 213-214 fix this up, see new line 220
Line 234 this line runs into the margin, see new lines 240-242
Line 264 ``According to the" , see new line 270
Line 266 ``we", see new line 272
Line 291 fix this up, see new lines 294-295
Lines 332-333 use more sentences here, see new line 333-334
Line 345 this line rums into the margin, see new lines 346-348
Line 352 replace ``by" by ``since", see new line 353
Lines 364-365 fix this up, see new line 364
Line 367 ``We omit the remainder of the proof", see new line 367
Line 385 this line rums into the margin, see new lines 385-386
Line 386 ``the fractional", see new line 387
Throughout the references you should put a space for example in line 404 you should have a space before ``R." We do as the request, see the reference.
Author Response File: Author Response.pdf
Reviewer 4 Report
1. On page 3, clarify what kind of fractional operators could be obtained from Def.4.
2. Include some references for the definitions presented in the paper.
3. When mention a function, write f and not f(x). E.g. on Lemma 6, write u\in...and not u(t)\in...
4. Several works on psi-FDE are not mentioned in the paper e.g.
Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Meth. Appl. Sci. (41), 336-352 (2018)
or
On systems of fractional differential equations with the psi-Caputo derivative and their applications. Math Meth Appl Sci. (in press)
Please cite these and other references on similar subject.
Author Response
On page 3, clarify what kind of fractional operators could be obtained from Def.4.
We add remark 1 as the clarification, see new lines 108-109.
2. Include some references for the definitions presented in the paper.
We list all the references for the definitions, except the one we defined ourselves.
3. When mention a function, write f and not f(x). E.g. on Lemma 6, write u\in...and not u(t)\in...
We make a change, please see new line 185
4. Several works on psi-FDE are not mentioned in the paper e.g.
Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Meth. Appl. Sci. (41), 336-352 (2018)
or
On systems of fractional differential equations with the psi-Caputo derivative and their applications. Math Meth Appl Sci. (in press)
We add these two references as [7] and [8], and we add two other papers as new references [9] and [26]
Author Response File: Author Response.pdf
