【作者】 厉亚；

【导师】 黄立宏；

【作者基本信息】 湖南大学 ， 数学， 2003， 硕士

【摘要】 本文主要研究下述二、三阶非线性泛函微分方程和解的振动性、渐近性与周期性。在σ既可以为奇数／奇数又可以为偶数／奇数的情况下，对方程(0.1)的振动性与渐近性获得了一系列新的充分判据。对具有多时滞的二、三阶方程(0.2)和(0.3)建立了若干新的周期解的存在性判据，改进或推广了已有文献中的一些相关结论。全文由四章组成，具体结构如下： 第1章主要介绍了问题研究的背景及所需要的预备知识。 第2章讨论了二阶非线性泛函微分方程(0.1)解的振动性与渐近性，推广了许多已有结果。 第3章研究了二阶非线性泛函微分方程(0.2)的周期解的存在性，通过Mawhin定理，获得了(0.2)周期解存在的若干新的充分条件，推广了最近文献[29]中的结果。 第4章考虑了三阶泛函微分方程(0.3)的周期解的存在性，参考最近文献[32]，利用新的分析技巧估计Mawhin延拓定理中集合Ω的先验界，得到了方程(0.3)周期解存在的若干充分性判据，改进和扩展了已有文献中的相关结论。更多还原

【Abstract】 In this paper, we consider oscillatory, asymptotic behavior of solutions and the existence of periodic solutions of the following second and third order nonlinear functional differential equationsSome new sufficient criteria are obtained to characterize oscillatory and asymptotic behavior of solutions of equation (0.1), when o is either the quotient of two odd numbers or one even and one odd numbers. In addition, we establish some existence criteria about periodic solutions of the second and third order equations (0.2) and (0.3) with delays. These results improve and generalize some existing conclusions in the literature. This paper consists of four chapters and is organized as follows:In Chapter 1, we introduce the background and some necessary preliminary knowledge .In Chapter 2, we discuss the oscillatory and asymptotic behavior of solutions of the second order nonlinear differential equation (0.1) and generalize some existing results.In Chapter 3, we study the existence of the periodic solutions of the second order nonlinear differential equation (0.2). By means of Mawhin theorem, we obtain some new sufficient conditions ensuring the existence of the periodic solutions of equation (0.2), and generalize the results obtained in the recent paper [29].In Chapter 4, we consider the existence of periodic solutions of the third order functional differential equation (0.3). By using the recent reference [32] and new analytical technique to estimate the prime bound of J.Mawhin theory, we obtain some sufficient criteria for the existence of periodic solutions, which improve some relative conclusions in the existing references.更多还原