This research focuses on studying the asymptotic and oscillatory behavior of a special class of even-order nonlinear neutral differential equations, including damping terms. The research aims to achieve qualitative progress in understanding the relationship between the solutions of these equations and their associated functions. Leveraging the symmetry between positive and negative solutions simplifies the derivation of criteria that ensure the oscillation of all solutions. Using precise techniques such as the Riccati method and comparison methods, innovative criteria are developed that guarantee the oscillation of all the solutions of the studied equations. The study provides new conditions and effective analytical tools that contribute to deepening the theoretical understanding and expanding the practical applications of these systems. Based on solid scientific foundations and previous studies, the research concludes with the presentation of examples that illustrate the practical impact of the results, highlighting the theoretical value of research in the field of neutral differential equations.

In this work, we investigate the oscillatory properties of the neutral differential equation ${(r(l){\left[{(s(l)+p(l)s(g(l)))}^{\prime }\right]}^{\mathfrak{v}})}^{\prime }+{\sum }_{i=1}^n{q}_i(l)s^{\mathfrak{v}}(h_i(l))=0$, where $s\ge s_0$. We first present new monotonic properties for the solutions of this equation, and these properties are characterized by an iterative nature. Using these new properties, we obtain new oscillation conditions that guarantee that all solutions are oscillate. Our results are a complement and extension to the relevant results in the literature. We test the significance of the results by applying them to special cases of the studied equation.

This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings.

This research deals with the study of the oscillatory behavior of solutions of second-order differential equations containing neutral conditions, both in sublinear and superlinear terms, with a focus on the noncanonical case. The research provides a careful analysis of the monotonic properties of solutions and their derivatives, paving the way for a deeper understanding of this complex behavior. The research is particularly significant as it extends the scope of previous studies by addressing more complex forms of neutral differential equations. Using the linearization technique, strict conditions are developed that exclude the existence of positive solutions, which allows the formulation of innovative criteria for determining the oscillatory behavior of the studied equations. This research highlights the theoretical and applied aspects of this mathematical phenomenon, which contributes to enhancing the scientific understanding of differential equations with neutral conditions. To demonstrate the effectiveness of the results, the research includes two illustrative examples that prove the validity and importance of the proposed methodology. This work represents a qualitative addition to the mathematical literature, as it lays new foundations and opens horizons for future studies in this vital field.

The main objective of our paper is to investigate the oscillatory properties of solutions of differential equations of neutral type and in the noncanonical case. We follow an approach that simplifies and extends the related previous results. Our results are an extension and reflection of developments in the study of second-order equations. We also derive criteria for improving conditions that exclude the decreasing positive solutions of the considered equation.

This paper explores the asymptotic and oscillatory properties of a class of third-order neutral differential equations with multiple delays in a non-canonical form. The main objective is to simplify the non-canonical form by converting it to a canonical form, which reduces the complexity of the possible cases of positive solutions and their derivatives from four cases in the non-canonical form to only two cases in the canonical form, which facilitates the process of inference and development of results. New criteria are provided that exclude the existence of positive solutions or Kneser-type solutions for this class of equations. New criteria that guarantee the oscillatory behavior of all solutions that satisfy the conditions imposed on the studied equation are also derived. This work makes a qualitative contribution to the development of previous studies in the field of neutral differential equations, as it provides new insights into the oscillatory behavior of neutral equations with multiple delays. To confirm the strength and effectiveness of the results, three examples are included that highlight the accuracy of the derived criteria and their practical applicability, which enhances the value of this research and expands the scope of its use in the field.