泛函微分方程周期解、同宿解及相关问题的研究

项目名称： 泛函微分方程周期解、同宿解及相关问题的研究

项目编号： No.11271197

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 鲁世平

作者单位： 南京信息工程大学

项目金额： 60万元

中文摘要： 本项目拟开展泛函微分方程周期解、同宿解及相关问题的研究。首先利用算子理论和差分方程理论研究非线性D-算子对中立型方程周期解先验界、可微性等影响的机理，在此基础上，利用凝聚场和非紧性测度场的拓扑度理论研究D-算子为非线性的中立型泛函微分方程周期解存在性、唯一性和多解性问题，设法免除已有相关研究中需Jack Hale提出的D-算子稳定的关键条件, 并探讨滞量对周期解性质影响的信息。其次，利用拓扑度理论，研究泛函微分方程同宿解的存在性、唯一性和多解性问题。通过探讨滞量对同宿解先验界的影响，获得同宿解存在性、唯一性和多解性与滞量之间的关系，从而揭示滞量对同宿解影响的信息。通过研究中立型偏泛函微分方程解的性质，给出解的长时间形态，揭示滞量与解的性质、空间扩散的关系。在此基础上，进一步研究具扩散的中立型种群模型周期正解的存在性、唯一性和多解性等问题。

中文关键词： 泛函微分方程；周期解；同宿解；拓扑度；种群模型

英文摘要： The plan for this project is to study the problems of periodic solution, homoclinic solution and it's related topics for functional differential equation. By using the theory of operator and difference equation, we investigate the mechanism under which how nonlinear operator D influences a priori bounds and the differentiability of periodic solutions for neutral functional differential equation in the first. Then, by means of the theory of topological degree associated condensing field and k-set contraction field, we study the existence, uniqueness and multiplicity of periodic solution for neutral functional differential equation with nonlinear operator D. The aim of this study is to generalize or withdraw the assumption that the operator D is stable, which is proposed by Jack Hale and has been crucial for studying the existence of periodic solution for neutral functional differential equation in the known literature, at same time, we try to investigate the message of how the delay reflects some general properties of the periodic solution. The second problem studied in this project is the existence, uniqueness and multiplicity of homoclinic solution for functional differential equations. In order to do it, we employ the theory of topological degree. Furthermore, through studying the relati

英文关键词： Functional differential equation；periodic solution；homoclinic solution；topological degree；population model