This work focuses on the canonical scenario and examines the oscillatory and asymptotic features of fourth-order differential equations with numerous delays and mixed neutral terms. The Riccati methodology is employed as a useful mathematical tool to simplify the theoretical analysis and derive stringent conditions that rule out the existence of positive solutions satisfying the examined equation. By systematically combining these conditions, precise criteria ensuring the oscillation of all solutions are obtained. These findings contribute qualitatively to the scientific literature by advancing the theoretical understanding of the oscillatory behavior of such equations. Furthermore, to highlight the practical importance of the established results, two applied examples are provided to demonstrate the effectiveness of the derived criteria in handling relevant mathematical models.

This research aims to introduce new criteria that guarantee the oscillation of all solutions to third-order nonlinear neutral differential equations in their canonical form. The proposed methodology integrates the comparison principle with first-order differential equations and employs the Riccati substitution technique, which simplifies the complex structure of the equations and transforms them into more analyzable forms. This approach contributes to establishing general and precise oscillation conditions, representing an extension and improvement over previously published work in this field. It is important to note that this study is purely analytical, focusing on the derivation of oscillatory properties and theoretical criteria. To validate the applicability of the results, three numerical examples are provided, demonstrating the capability of the proposed criteria to verify the oscillation of solutions and highlighting both the theoretical and practical significance of the methodology.

This paper investigates novel sufficient conditions for the oscillatory behavior of fourth-order nonlinear differential equations with mixed neutral terms. By employing refined Riccati transformation techniques and advanced analytical approaches, we establish extended criteria that enrich the theoretical understanding of oscillation phenomena within this class of neutral differential equations. The proposed results significantly improve upon previously known conditions in the literature. Moreover, illustrative numerical examples are provided to demonstrate the applicability and sharpness of the obtained criteria. The findings contribute to the broader framework of nonlinear analysis and offer valuable insights into the oscillatory dynamics of functional differential equations with delay and neutral terms.

This article explored oscillation conditions for quasi-linear neutral differential equations of noncanonical form. By establishing new iterative monotonic properties of solutions, we derived novel oscillation criteria that extend and refine existing results in the literature. To illustrate the significance of our findings, we provided three concrete examples demonstrating the applicability of the proposed conditions.