非线性微分方程的奇异边值问题与周期解分支

项目名称： 非线性微分方程的奇异边值问题与周期解分支

项目编号： No.10871063

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 罗治国

作者单位： 湖南师范大学

项目金额： 26万元

中文摘要： 本项目研究非线性微分方程的动力学性态。研究了一阶、二阶泛函微分方程和脉冲微分方程边值问题，获得了这些系统的正解或极值解的新的存在性定理；讨论了无穷区间上二阶脉冲奇异边值问题，给出了这类系统的多解存在的条件；利用临界点理论和变分方法研究了一些二阶脉冲微分方程非线性边值问题的多解性；研究了一类二阶脉冲微分方程的极小周期问题，给出了这些系统的次调和解存在的充分条件；建立了脉冲泛函微分方程零解一致稳定和一致渐近稳定的新的判据；研究了一类具Lipschitz激励功能的双向联想记忆(BAM)神经网络模型，获得了该模型的唯一平衡点的存在性和渐近稳定性的条件；我们还利用k-集压缩算子的抽象延拓定理研究了一类多时滞对数种群模型的正周期解的存在性，获得了新的存在性定理。

中文关键词： 非线性微分方程；边值问题；周期解；稳定性

英文摘要： This project researched the dynamical properties of nonlinear differential equations. We investigated the boundary value problems for first and second order functional differential equations and impulsive differential equations, proved new existence theorems of positive solutions or extreme solutions for these systems; We studed the singular boundary value problems for second order impulsive differential equations on the half line, The conditions for the existence of multiple positive solutions are established; By use critical point theory and variational methods, we discussed the existence of multiple solutions to second order boundary value problems with impulsive effects; We investigated the subharmonic solutions with prescribed minimal period for a class of second order impulsive differential equations, The conditions for the existence of subharmonic solutions are established; We obtained new uniformly stability theorems and uniformly asymptotic stability theorems for impulsive functional differential equations; Sufficient conditions are obtained for the existence and asymptotic stability of a unique equilibrium of a Bidirectional Associative Memory (BAM) neutral network with Lipschitzian activation functions; We used the abstract continuous theorem of k-set contractive operator to dicuss a neutral multi-delay logarithmic population model. a new result is obtained for the existence of positive periodic solutions to this system.

英文关键词： Nonlinear differential equation； Boundary value problem；Periodic solution；stability