This work investigated the asymptotic performance of nonoscillatory solutions to the functional differential equation (FDE) ${\left(\alpha \left(u\right){\left|y^{\mathrm{\prime }}\left(u\right)\right|}^{\kappa -1}{y}^{\mathrm{\prime }}\left(u\right)\right)}^{\mathrm{\prime }}+p\left(u\right)F\left(y^{\mathrm{\prime }}\left(\delta \left(u\right)\right)\right)+q\left(u\right)G\left(y\left(\rho \left(u\right)\right)\right)=0$, which involves a delayed damping term. Using Riccati and comparison methods, we extended the previous results to the nonlinear case of the considered equation. Furthermore, the new criteria improved upon the previous ones by removing some constraints on the delay functions. Then, for the linear case, we derived new criteria that take into account all parameters of the equation. The examples and comparisons provided illustrate the importance and novelty of our results.

This study investigates the oscillatory properties of solutions for a general class of neutral differential equations with multiple delays. Using Riccati and comparison techniques, we establish five distinct oscillation theorems that address the limitations of previous results in this topic. Our criteria not only extend and generalize earlier findings but also reduce the required constraints. Notably, they provide sharper results when applied to special cases like Euler's equation. The novelty and effectiveness of the proposed oscillation criteria are illustrated through a detailed analysis of a given example, supported by tables and figures.

This article delves into the behavior of solutions to a general class of functional differential equations that contain a neutral delay argument. This category encompasses the half-linear case and the multiple-delay case of neutral equations. The motivation to study this type of equation lies not only in the exciting analytical issues it presents but also in its numerous vital applications in physics and biology. We improved some of the inequalities that play a crucial role in developing the oscillation test. Then, we used an improved technique to derive several criteria that ensure the oscillation of the solutions of the studied equation. Additionally, we established a criterion that did not require imposing monotonic constraints on the delay functions and took into account their effect. We have supported the novelty and effectiveness of the results by analyzing and comparing them with previous results in the literature.

In this study, we establish new oscillation criteria for the fourth-order delay differential equation. Our main objective is to build upon recent advancements in the study of the oscillatory behavior of second-order equations and extend these findings to higher-order equations. Although fourth-order equations have numerous applications, their study presents significant analytical challenges due to the complex nature of their solutions, which we will discuss in this study. We use the comparison technique with first-order equations in several approaches. Our results show an improvement in the oscillation test due to the development of some monotonic and asymptotic properties of positive solutions. We present a comparison of the new criteria to test their effectiveness, as well as a comparison with previous studies to illustrate the novelty.

This research aimed to find numerical solutions to a type of nonlinear initial value problem (IVP) for hybrid fractional differential equations. Using the Adomian decomposition method (ADM) and the Picard method (PM), we studied the Chandrasekhar quadratic integral equation (QIE). Furthermore, we investigated existence and uniqueness results using measures of weak noncompactness. Through a set of examples and numerical simulations, a comparison was made between the results of the AMD and PM.

The study of the oscillatory behavior of a general class of neutral Emden-Fowler differential equations is the focus of this work. The main motivations for studying the oscillatory behavior of neutral equations are their many applications as well as the richness of these equations with exciting analytical issues. We obtained novel oscillation conditions in Kamenev-type criteria for the considered equation in the canonical case. We improve the monotonic and asymptotic characteristics of the non-oscillatory solutions to the considered equation and then utilize these characteristics to refine the oscillation conditions. We present, through examples and discussions, what demonstrates the novelty and efficiency of the results compared to previous relevant findings in the literature. In addition, we numerically represent the solutions of some special cases to support the theoretical results.