Oscillatory Analysis of Third-Order Hybrid Trinomial Delay Differential Equations via Binomial Transform

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript presents a novel approach to analyzing the oscillatory behavior of a class of third-order hybrid-type delay differential equations. The proposed method relies on transforming the original trinomial equation into a simplified binomial form. To establish the oscillatory nature of the solutions, the authors employ comparison techniques and integral averaging methods. An illustrative example is provided to support the theoretical results.

The topic addressed is relevant, the methodology is rigorous, and the paper is well-structured, with a clear progression from the problem formulation to the theoretical developments and their application. However, before the manuscript can be considered for publication, the following points should be addressed:

1) The quality of the English language requires improvement. Typographical errors should be corrected, and the use of informal expressions such as “one may call…” or “we use…” should be avoided in favor of more formal alternatives.

2) The degree of novelty in comparison to existing literature should be more clearly articulated. A more detailed discussion on this aspect would significantly enhance the manuscript's contribution.

3) The conclusion is currently too concise. It is recommended to expand this section by explicitly summarizing the implications of the main theorems, outlining potential extensions of the work, and discussing possible practical applications.

4) Section 4 could benefit from the inclusion of a graphical representation of an oscillatory solution, which would provide a valuable visual complement to the theoretical analysis.

Author Response

Please refer to the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper investigated the oscillatory behavior of a specific group of third-order functional differential equations that feature both delay and hybrid characteristics, incorporating both positive and negative terms. The authors introduce an innovative analytical approach by converting the original complex trinomial equations into a binomial format through the use of the binomial transform, alongside auxiliary linear differential equations. This transformation allows for the implementation of comparison and integral averaging techniques, leading to the development of new and precise oscillation criteria. The study thoroughly establishes sufficient conditions that ensure all solutions will oscillate, offering a richer insight into the structure and dynamics of these differential equations. The findings not only build upon and refine existing theories but are also backed by a comprehensive example that demonstrates the relevance and importance of the proposed methods.

 

The topic of the paper is interesting and worthy of investigation. I have carefully read the paper and believe that it can be published after some major modifications as follows:

 

- What is the application of "Third-Order Hybrid Trinomial Delay Differential Equations " in real world? The introduction section should be enriched with such discussions and more introduction to this specific model.

- Why is it needed to study the oscillatory and nonoscillatory conditions of the problem? Its advantageous and benefits are not clarified well?

- The new contribution of this study is not explained in detail.

- What are the assumptions and limitations of your models as well as the restrictions of the Binomial Transform method?

- It is suggested to add some illustrative examples to show the dynamic behavior of the Hybrid Trinomial Delay Differential Equations.

- The literature review is not accurate enough to reflect the importance and show the real applications of the considered problem. It is highly recommended to enrich this section by discussing more relevant works regarding "Dynamics of Delay Differential Equations". The following articles in the literature are suggested to be included and discussed. "On the Dynamics of the Logistic Delay Differential Equation with ‎Two Different Delays"

- More discussions on the results are needed to present the major outcomes of the present study.

- The conclusion section should be rearranged to express the main concluding remarks of the present work.

- The paper should be double-checked from a grammatical point of view.

 

Author Response

Please refer to the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Line 19: " $\alpha$ is a ratio of the old positive integers," what are the "old" integers? Probably, odd?

Line 22: "$C′([t_0,\infty),\mathbb{R})$" What is the space? Continuously differentiable functions?

The motivation and actuality part in Lines 32-33, 42-48 is very weak and unclearly written.

The results are interesting. But they are very carelessly written and formulated. 

The authors should strengthen their results by at least considering the case of a complex parameter $t\in\mathbb{C}$. Then it is necessary to consider the oscillatory nature of the solutions along the every fixed complex ray from the origin.

For equation (19) the authors must indicate explicit form of at least one oscillatory solution (exact form).

Definition 1 depends on the y_2. Given this, the notion of principal solution is unclearly posedness because for some y_2 the solution y_1 is principal and fo some y_2 the solution y_1 is not principal.

The presented results and their proofs are very simple, and their significance do no achieve high scientific level  of the journal Mathematics.

Comments on the Quality of English Language

Line 23. "satisfies (1) for $t \ge t_0$ and satisfies (1) for $t \ge t_0$."

Line 33. "where positive solution". Probably, "where" must be replaced by "with"

P.4. Line after Equation (12) "Using (9) in (12) yields" must be replaced by "Using (9) in (12) it yields"

 

Author Response

Please refer to the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Accept as is.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have Implemented completely my remarks.