二阶奇异微分方程周期解与无界解的存在性研究

Singular differential equation is a kind of equations with strong practical background. Many of its models originate from some practical problems, such as the Brillouin electron beam focusing equation and the Emarkov-Pinney equation. This project focuses on the theories of singular differential Equations and their applications: by using phase plane analysis and topological degree, we study the existence and multiplicity of periodic solutions for the various resonance cases; we study the relation between periodic potential and resonance and non resonance, study the existence and multiplicity of periodic solutions of singular differential equations with periodic potential by using phase plane analysis and topological degree or using Variational method, and study the applications to the Brillouin electron beam focusing equation and the Emarkov-Pinney equation; we study the existence of unbounded solutions of singular differential equations by using phase plane analysis, and study the co existence of unbounded solutions and periodic solutions. These questions are very important theoretical and practical issues in the field of singular differential equations, so the research in this project has great theoretical and practical value, and can further improve the relevant theory of singular differential equation.

奇异微分方程是一类具有很强实际背景的方程，它的很多模型来源于一些实际问题，比如电子聚焦束方程问题和Emarkov-Pinney方程。本项目着重研究奇异微分方程中的下列理论与应用问题：利用相平面分析和拓扑度或者扭转定理研究各种共振情形下周期解的存在性和多解性；研究周期位势与共振和非共振的本质关系，利用相平面分析和拓扑度或者利用变分法研究带周期位势的奇异微分方程周期解的存在性与多解性，并利用所得到的结果研究电子聚焦束方程和Ermakov-Pinney方程；利用相平面分析研究各种共振形式下奇异微分方程无界解的存在性，并研究无界解和周期解的共存性。这些问题都是奇异微分方程领域非常重要的理论和应用问题，故本课题的研究具有非常好的理论价值和实际应用价值，并可进一步完善奇异微分方程的相关理论体系。

奇异微分方程是一类源于很强实际背景的方程，研究它的诸多问题具有很好的理论和实际意义。本项目研究二阶奇异微分方程的周期解和无界解问题，探讨电子聚焦束方程问题和Emarkov-Pinney方程周期解的存在性和多解性等问题。我们找到一种新的寻找周期解的方法，并经过理论研究和计算机模拟计算发现，致使周期解的存在的参数的范围严重依赖奇异项的次数，并随次数的变化而变化。这为研究一般形式的带周期位势的奇异微分方程提供了一种方法参考，同时在实际应用中为研究电子聚焦束方程和Emarkov-Pinney方程等提供了参考方法。