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

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

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Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

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{\bf Referee's report} 

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

Method and Arithmetic-Geometric Inequality"}  

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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 $z(t) = x(t) + bx(t-\sigma).$ 

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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). 

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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 $b_1(t) = 1 = \gamma_1$. 

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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 $b_1(t) \ne 1 \ne \gamma_1$.

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Comments for author File: Comments.pdf

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Reviewer 3 Report

Comments and Suggestions for Authors

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Comments for author File: Comments.pdf

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Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

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Comments for author File: Comments.pdf

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Round 3

Reviewer 3 Report

Comments and Suggestions for Authors

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