In this work, we study the asymptotic and oscillatory behavior of solutions to the second-order general neutral Emden–Fowler differential equation ${\left(a\left(v\right)\eta \left(x\left(v\right)\right)z^{\prime }\left(v\right)\right)}^{\prime }$ + $q\left(v\right)F\left(x\left(g\left(v\right)\right)\right)$ = 0, where $v\ge v_0$ and the corresponding function z = x + p$\left(x\circ h\right)$. Besides the importance of equations of the neutral type, studying the qualitative behavior of solutions to these equations is rich in analytical points and interesting issues. We begin by finding the monotonic features of positive solutions. The new properties contribute to obtaining new and improved relationships between x and z for use in studying oscillatory behavior. We present new conditions that exclude the existence of positive solutions to the examined equation, and then we establish oscillation criteria through the symmetry property between non-oscillatory solutions. We use the generalized Riccati substitution method, which enables us to apply the results to a larger area than the special cases of the considered equation. The new results essentially improve and extend previous results in the literature. We support this claim by applying the results to an example and comparing them with previous findings. Moreover, the reduction of our results to Euler’s differential equation introduces the well-known sharp oscillation criterion.

The goal of this study is to derive new conditions that improve the testing of the oscillatory and asymptotic features of fourth-order differential equations with an advanced neutral term. By using Riccati techniques and comparison with lower-order equations, we establish new criteria that verify the absence of positive solutions and, consequently, the oscillation of all solutions to the investigated equation. Using our results to analyze a few specific instances of the examined equation, we can ultimately clarify the significance of the new inequalities. Our results are an extension of previous results that considered equations with a neutral delay term and also an improvement of previous results that considered only equations with an advanced neutral term.

This article explores the oscillation properties of solutions for higher-order nonlinear neutral functional differential equations. This study sought to establish applicable oscillation conditions for nonlinear NDDEs when $\alpha \ne \beta$. Our analysis focused on identifying the key asymptotic characteristics of positive solutions in the noncanonical case. Based on these properties, we developed a multi-conditional criterion that guarantees oscillation, thereby extending previously established results for second-order equations. Unlike previous results, which involved arbitrary constants and were thus difficult to apply, the criteria derived in this work do not contain such constants, making them directly applicable. Moreover, to overcome the complexity introduced by the multiple constraints of this criterion, we propose a new and previously unreported simplified oscillation condition. The effectiveness of this simplified criterion is demonstrated through an illustrative example.