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Peer-Review Record

Oscillation and Asymptotic Behavior of Third-Order Neutral Delay Differential Equations with Mixed Nonlinearities

Mathematics 2025, 13(5), 783; https://doi.org/10.3390/math13050783
by Balakrishnan Sudha 1, George E. Chatzarakis 2,* and Ethiraju Thandapani 3
Reviewer 1:
Reviewer 2: Anonymous
Reviewer 3: Anonymous
Mathematics 2025, 13(5), 783; https://doi.org/10.3390/math13050783
Submission received: 19 January 2025 / Revised: 21 February 2025 / Accepted: 24 February 2025 / Published: 27 February 2025
(This article belongs to the Section C1: Difference and Differential Equations)

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper contains new oscillation results for third-order neutral delay differential equations. The paper is carefully written, results are interesting and illustrated by some examples.

Author Response

Please see the attachment

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

\begin{document} 

\pagestyle{empty}

\begin{center} 

{\bf Referee's report} 

\end{center} 

\begin{center} 

{\bf "Third-Order Neutral Delay Differential Equations with Mixed Nonlinearities: Almost Oscillation via Linearization

Method and Arithmetic-Geometric Inequality"}  

\end{center} 

 

In this paper, the authors investigate third-order nonlinear neutral delay differential equations with

mixed nonlinearities of the form

$$

(1) \;\; (a(t)(z{''}(t))^{\alpha})' + \sum_{i=1}^n p_i(t) x^{\alpha_i}(\tau_i(t)) = 0, \;\; t \geq t_0 \geq 0, 

$$

where $z(t) = x(t) + bx(t-\sigma).$ 

\medskip

 

The main result of this work provides sufficient conditions for the oscillatory and asymptotic behavior of the solutions of (1). By means of a linearization method and an arithmetic-geometric inequality, the authors obtain some new criteria for the oscillatory and asymptotic behavior of the solutions of (1). 

\medskip

 

This work is motivated by papers [21] and [22]. In particular, it is mentioned that the results published in [22] are not entirely correct. In [22], the more general equation is studied.

$$

(2) \;\; (b_2(t)((b_1(t) (z'(t))^{\gamma_1})')^{\gamma_2})' + \sum_{i=1}^n q_i(t) x^{\alpha_i}(\tau_i(t)) = 0, \;\; t \geq t_0 \geq 0. 

$$

 

The paper under review appears to be correct and well-written and corrected the results in [22], at least for $b_1(t) = 1 = \gamma_1$. 

\medskip

 

The manuscript under review can be accepted for publication, however it would be interesting if the authors could comment on how they can extend their results for the equation (2) when $b_1(t) \ne 1 \ne \gamma_1$.

 

\end{document} 

 

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.docx

Round 3

Reviewer 3 Report

Comments and Suggestions for Authors

.

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