Uniqueness of Solution for Impulsive Evolution Equation in Ordered Banach Spaces^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

In this manuscript, the authors study the existence and uniqueness of mild
solution for semilinear impulsive evolution equation in an ordered Banach space. Without the assumption of upper and lower solutions, they firstly prove that the corresponding
linear periodic boundary value problem has a unique positive solution $w_0$ by using the Poincar\'{e} operator and contraction mapping principle and using the monotone iterative method in $[-w_0,w_0]$, they obtain the existence and uniqueness of mild solutions to impulsive evolution equation. In my opinion, the paper has innovative points and uses a good method to prove the uniqueness of mild solutions to impulsive equations. According to the previous comments, I recommend its publication in ``Mathematics'' but the revised version.
\begin{enumerate}
\item {Page 2, line 24, 30, 38: ``the" should be deleted.}

\item {Page 3, line 56: ``Furthermore,$\cdots$ PBVP(1.1)", We didn't find the relevant result in the text.}

\item {Page 4, line 81: ``linear periodic bounded value problem" should be ``linear periodic boundary value problem".}

\item {Page 4, Lemma 2.1: ``generates" should be ``generate".}

\item {Page 8, line 112: ``an" should be ``a".}

\item {Page 4,9, The range of $h$ in Theorem 3.1 is inconsistent with that in Lemma 2.1. Please explain that.}

\item {Page 10, condition (H1): ``$f(t,-x)\geq -ax-h(t)$" should be ``$f(t,-x)\geq -ax+h(t)$" and ``$I_k(-x)\geq -a_kx-b_k$" should be ``$I_k(-x)\geq -a_kx+b_k$".}

\item {Page 10, The proof process in Theorem 3.1 needs to be further improved.}

\item {Page 13, line 169: ``$A+CI$" what is $C$?.}

\item {Page 14, line 170: There is less $\leq$ in the formula.}

\item {Page 15, line 180: ``derivatives" should be ``derivative".}

\item {Page 15, Theorem 4.1 contains the positive solution, and Theorem 3.2 is the unique mild solution. Please supplement or correct it.}
\end{enumerate}

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper, the authors study a periodic boundary value problem of impulsive evolution equation in an ordered Banach space. The results seem correct, but the manuscript should be improved by making some changes. For example:

Line 20: ``In`` should be ``in``

Line 38: ``the`` instead of ``The``

Line 106: ``, For`` should be ``, for``

Line 130: ``Then`` should be ``then`` and ``,`` instead of ``.`` at the end of the previous line

Line 178: ``assume`` instead of ``assuming``

In the second line of the proof of Theorem 1: replace ``And`` with ``We``

Between lines 89 and 90: ``... It is well-known [17, Chapter 4, Theorem 2.9] that for any ..., the unique solution of initial value problem (2.3) is given by ..., then ...``. It doesn’t make sense!

Maybe it would be good for the authors to add, for example, the definitions of Poincare mapping, resolvent operator,  to make the article understandable for a wider class of readers.

General remark//All over the paper

Between name of the author(s) and [cited paper] should be a space.

 

I believe that the manuscript cannot be published in its current form, but only after the authors intervene and improve it.

Comments on the Quality of English Language

through the corrections above there are also those related to the English Language

Author Response

In this paper, the authors study a periodic boundary value problem of impulsive evolution equation in an ordered Banach space. The results seem correct, but the manuscript should be improved by making some changes. For example:

Line 20: ``In`` should be ``in``

Line 38: ``the`` instead of ``The``

Line 106: ``, For`` should be ``, for``

Line 130: ``Then`` should be ``then`` and ``,`` instead of ``.`` at the end of the previous line

Line 178: ``assume`` instead of ``assuming``

In the second line of the proof of Theorem 1: replace ``And`` with ``We``

Between lines 89 and 90: ``... It is well-known [17, Chapter 4, Theorem 2.9] that for any ..., the unique solution of initial value problem (2.3) is given by ..., then ...``. It doesn’t make sense!

Maybe it would be good for the authors to add, for example, the definitions of Poincare mapping, resolvent operator,  to make the article understandable for a wider class of readers.

General remark//All over the paper

Between name of the author(s) and [cited paper] should be a space.

 

I believe that the manuscript cannot be published in its current form, but only after the authors intervene and improve it.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

See the attached report

Comments for author File: Comments.pdf

Author Response

The manuscript investigates the periodic boundary value problem (PBVP) for impulsive evolution equations in ordered Banach spaces. By employing Poincar\'{e} mapping techniques and the
monotone iterative method, the authors prove:
\begin{itemize}
  \item { The existence of mild and positive mild solutions to PBVPs without assuming pre-defined
upper and lower solutions.}
  \item { The uniqueness of mild solutions using resolvent spectral radius estimates.}
  \item { An application to impulsive parabolic PDEs demonstrating the theory's applicability. The paper contributes meaningfully to the theory of evolution equations, especially in the context of impulse effects and periodic behavior.}
\end{itemize}
Comments/Suggestions:
\begin{enumerate}
  \item { Hypotheses (H1)-(H6) should be more clearly motivated or illustrated. Add motivating
examples and interpretations for each condition, especially how they control the behavior of
$f$ and $I_k$.}
 \item { The use of Arzel\`{a}-Ascoli and compactness in Section 3.1 would benefit from more narrative
context or schematic diagrams.}
 \item { Numerous grammatical errors are present (e.g., ``constracted'' should be ``constructed'', ``Latter'' should be ``Later''). A comprehensive proofreading is necessary.}
 \item { The example in Section 4 is somewhat abstract; including concrete forms of $g(x,t,u)$ and
$\phi_k(x,u)$ would help.}
 \item { Define the mapping $\Pi$ explicitly and describe its role in proving existence via the contraction
mapping theorem. Provide more detail on how the periodicity condition translates into a
fixed-point equation. }
 \item { Moreover, the introduction should contain a short paragraph based on
recent applications of fixed-point theory, see for example:
Some Novel Proximal Point Results and Applications. Univ. J. Math. Appl. 2025;8(1):8-20.}
 \item { Summarize implications of spectral radius bounds (e.g., contraction implications) in plain
language where possible.}
 \item {  Example in Section 4: Strengthen this section with explicit choices for $g(x,t,u)$ and $\phi_k(u)$.
For example, take
$$g(x,t,u)=\frac{u}{1+u^2}+\sin(t),\,\phi_k(u)=\frac{u}{1+|u|}$$
and verify that they satisfy the required assumptions.}
 \item { Mild Solution Definition: Reintroduce the definition of a mild solution using the variation of
constants formula to reinforce clarity.}
\end{enumerate}

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

Not all of the corrections indicated are made. In the new version, in the first line of page 2, the first red text is the old one, not the one that should have been corrected. The same happens with several corrections indicated. In addition, in the new parts, written in red, there are also new mistakes. 
I do not consider that the corrections were made, but, even, in addition, others appeared ... . The authors should carefully review the entire manuscript according to the old indications (applied also to the new parts) .
I do not recommend the publication of this version. 

Comments on the Quality of English Language

the old language corrections were not made
there are also new corrections: for example, on the 3rd line before the References, you written '... and ... is monotonically increasing' instead of '... and ... are monotonically increasing'.

Author Response

Dear Authors,

Not all of the corrections indicated are made. In the new version, in the first line of page 2, the first red text is the old one, not the one that should have been corrected. The same happens with several corrections indicated. In addition, in the new parts, written in red, there are also new mistakes. 
I do not consider that the corrections were made, but, even, in addition, others appeared ... . The authors should carefully review the entire manuscript according to the old indications (applied also to the new parts) .
I do not recommend the publication of this version. 

the old language corrections were not made
there are also new corrections: for example, on the 3rd line before the References, you written '... and ... is monotonically increasing' instead of '... and ... are monotonically increasing'.

Response:The mentioned content has been revised. Thank you for your correction.

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

 

Thank you for the new form of the manuscript.