一类n阶非线性泛函微分方程的振动性和渐近性
Oscillation and Asymptotic Properties of a Class of nth-Order Nonlinear Functional Di?erential Equations
摘要: 本文聚焦于带连续分布时滞的n阶非线性中立型泛函微分方程，深入研究其解的振动性与渐近性行为。通过运用推广的Riccati变换、Philos型积分平均方法以及Young不等式等数学工具，结合方程的结构特征与基本假设，推导出该类方程解的振动性判定新准则，同时得到非振动解的渐近收敛性质。所得结果推广并改进了现有针对三阶或偶数阶方程的研究结论，拓宽了高阶微分方程振动理论的适用范围。
Abstract: This paper focuses on the nth-order nonlinear neutral functional differential equa- tions with continuously distributed delays, and studies the oscillation and asymptotic behavior of their solutions in depth. By using the generalized Riccati transforma- tion, Philos-type in-tegral averaging method and Young’s inequality, combined with the structural characteristics and basic assumptions of the equation, new oscillation criteria for the solutions of this class of equations are derived, and the asymptotic convergence properties of non-oscillatory solutions are obtained. The results general- ize and improve the existing research conclusions only for third-order or even-order equations, and expand the application scope of the oscillation theory of higher-order functional differential equations.
文章引用：任勇, 郑诗巾. 一类n阶非线性泛函微分方程的振动性和渐近性[J]. 应用数学进展, 2026, 15(6): 87-101. https://doi.org/10.12677/AAM.2026.156269

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