拓扑方法与奇异微分方程周期解、拟周期解研究

Singular differential equations is widely applied in astronomy, physics,biology and many other physical phenomena. Such type of equations has high values not only in the research fields, but also in practice,and therefore have become a very important part in the fields of differential equations and dynamical systems. The aim of this project is to systematically study the existence of periodic solutions and quasi-periodic solutions, multiplicity for several classes of singular differential equations by using topological degree,cone theory,and lower and upper solution method. In particular,for the cases of semilinear and superlinear, we will establish and develop the existence periodic and quasi-periodic solutions to radially symmetric Keplerian-like systems, exploring its more complex dynamical behavior.The results obtained here complement the work due to Fonda. Meanwhile, we discuss both the case of weak singularities and the case of strong singularities, which play a different role linked to periodic solutions and quasi-periodic solutions singular differential equations. We try to understand the essential differences among them, overcoming the variational theory can only deal with the attractive singularity. Owing to the importance in both theory and application, we consider the properties of the Green function for fractional differential equations with periodic and anti-periodic boundary value problem. As an application of the Green function, we give some multiple positive solutions to singular problem by means of Leray-Schauder nonlinear alternative, partial order method,and so on. The target of this project is to initially form a research system with some characteristics.

奇异微分方程在天文、物理、生物等学科中有着广泛的应用，具有很高的学术价值和理论价值，是微分方程和动力系统领域中一个重要的研究课题。本项目旨在综合运用拓扑理论的各种方法，系统地研究几类奇异微分方程周期解、拟周期解的存在性与多重性。尤其是针对半线性、超线性等奇异情形，建立Kepler型径向对称系统周期轨、拟周期轨的存在性理论，探索其更加复杂的动力学行为，这是对意大利数学家Fonda工作的重要补充。同时探讨奇异微分方程其弱奇性和强奇性在周期解、拟周期解的存在性方面所发挥的不同作用，尝试澄清两类奇异性的本质不同，克服变分理论只能处理吸引奇异的局限性。由于分数阶微积分具有广泛的应用背景，还将细致研究分数阶微分方程周期、反周期边值问题的Green函数，并运用非线性分析理论给出奇异问题解的存在性结果。我们的目标是经过努力，初步形成有一定特色的研究思路和体系。

非线性泛函分析是分析数学中既有深刻理论又有广泛应用的研究学科，它以数学和自然科学中出现的非线性问题为背景，建立处理非线性问题的若干一般性理论和方法。研究非线性问题的方法主要有变分方法、半序方法、拓扑度方法、解析方法等。奇异微分方程出现在各种应用数学和物理学中，如：气体动力学、核物理、化学反应和原子结构的研究，是微分方程和动力系统领域中一个重要的研究课题。利用拓扑度理论和变分方法，我们研究了几类奇异微分方程（系统）周期解的存在性与多重性。尤其是针对半线性、超线性等奇异情形，建立开普勒型径向对称系统周期轨的存在性理论，所得到的结果是对意大利数学家Fonda工作的重要补充。 同时，我们探讨了弱奇性和强奇性在奇异微分方程（系统）周期解存在性方面所发挥的不同作用。