泛函微分方程周期解、同宿解及相关问题的研究

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 relation between the delay and a priori bounds of homoclinic solution, we investigate the mechanism under which how the delay influences the existence, uniqueness and multiplicity of homoclinic solution...Finally, through studying some properties of partial differential equation of neutral type, we plan to investigate how the solution varies over a long period time and, we also investigate the message of how the delay influences some properties of the solution, as well as, the message of how the delay reflects the diffusion of the solution on the space. On this basis, we further study the existence, uniqueness and multiplicity of positive periodic solution for neutral population model with diffusion.

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。

本项目主要开展了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了线性、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Mawhin重合度拓展定理研究了一类D-算子为线性的中立型泛函微分方程同宿解存在性问题。 进一步利用凝聚场和非紧性测度场的拓扑度理论研究了D-算子为非线性的中立型泛函微分方程周期解问题，免除了已有相关研究中需Jack Hale提出的D-算子稳定的关键条件。同时，我们还研究了一类具奇异的中立型泛函微分方程周期正解问题，在奇异项为排斥型，且具奇强奇性的条件下，获得了周期正解存在性的新结果。通过探讨滞量对方程解的动力学性质的影响，获得了周期解、同宿解存在性与滞量之间的关系，从而揭示滞量对周期解、同宿解影响的信息。利用临界点理论和变分法原理，我们还研究了一类四阶微分方程同宿解问题，获得了至少存在两个同宿解的结果，改进了已有文献中相关结论。在生物种群动力学研究中，我们建立了一类带自由边界的入侵物种扩散问题的数学模型，该模型考虑了时空和环境对种群生长的影响。利用自由边界刻画扩散对生态随时间的变化状态，揭示了入侵物种扩散规律。我们的结果表明：在有利的栖息环境下，如果扩散较慢或食物充沛，入侵物种是可以漫延的；在不利的栖息环境下，初始值较小的入侵物种将会灭绝。在研期间， 项目组成员共发表学术论文35篇，其中发表在《J. Differential Equations》， 《Discrete and Continuous Dynamical Systems-Series》, 《Acta Mathematica Sinica English Series》等SCI(含SCIE)期刊的论文25篇. 被SCIE期刊录用待发表论文3篇，出版专著一部. 我们的成果被他人引用40多次，其中项目组成员林支桂教授等的论文：The spreading front of invasive species in favorable habitat or unfavorable habitat，J. Differential Equations, 257（2014）145-166 被他人引用16次(见Web of science).