**5. Conclusions**

Our aim in this article was to extend the approach taken in [14] to neutral equations and also to the non-canonical case. The study of the non-canonical case contains more analytical difficulties than the canonical case due to the possibility of the existence of positive decreasing solutions.

We deduced some asymptotic and monotonic properties of the positive solutions whose corresponding function is in class P*s*. Then, we created new oscillation parameters depending on the inferred characteristics. In addition, we iteratively derived these properties, so that it allows them to be applied more than once in the case of failure at the beginning. The results obtained in this article are characterized by the fact that they do not require the existence of unknown functions, unlike the results in [26] that require this. In addition, our results do not need the conditions *h* ◦ *g* = *g* ◦ *h* and *h* is nondecreasing, which are necessary conditions for the results in [28].

Extending our results in this study to the nonlinear case of the investigated equation would be very interesting. This is due to many analytical difficulties that must be addressed to obtain improved monotonic properties in the nonlinear case.

**Author Contributions:** Conceptualization, B.A. and O.M.; methodology, B.A. and O.M.; investigation, B.A. and O.M.; writing—original draft preparation, B.A. and O.M.; writing—review and editing, B.A. and O.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number RI-44-0820.

**Conflicts of Interest:** The authors declare no conflict of interest.
