弱奇性排斥周期系统解的存在性与稳定性研究

The project is based on the theory of nonlinear analysis and applications to singular semi-positive differential equations. There are certain international research and research results have a high value, which are characteristic of this project. To solve the existence and stability of solutions to periodic singular differential equations are difficult nature of large, especially for semipositone problem. This project will be considered with the existence and stability of periodic solution to second-order singular semi positone differential equations. Further we extend to the second-order non-autonomous semi positive coupled systems with weak singularity. We get the existence of solutions by cleverly constructing the Green's function and Leray-Schauder degree theory. Another We get the stability of solutions by twist torsion factor theory. About half of singular and equations are differential stability studies are currently no such result , will have a better originality , but also for the weak singular singular integral equations of the existence of positive solutions for research has laid a solid.foundation. At present it is very little on the study of stability to singular semi positive differential systems. Therefore, this project will be very meaningful and innovative.

本项目立足于非线性泛函分析理论与应用的研究，以奇异半正微分方程为主要研究对象，研究内容有一定的国际前沿性，研究成果具有较高的应用价值，这是本项目的特色.由于奇异微分方程及方程组周期正解的存在性和稳定性等性质的研究较难较大，尤其是半正问题的研究成果较少.本项目将考虑具有弱奇性的二阶半正微分方程周期正解的存在性和稳定性，并将其推广到具有弱奇性的二阶非自治半正耦合方程组的研究中.我们将通过巧妙的构造格林函数并利用Leray-Schauder度理论和不动点定理的新技巧和新方法研究弱奇性的二阶半正微分方程周期正解的存在性.利用扭转系数理论构造扭转系数并通过估计其上下界研究方程及方程组周期正解的稳定性.关于奇异半正微分方程及方程组稳定性的研究目前没有此类结果，将具有较好的独创性,同时也为弱奇性奇异积分方程正解存在性的研究奠定了坚实的基础.

非线性奇异微分方程周期问题问题是方程理论中的重要课题，它的丰富理论和先进方法为解决当今科技领域中层出不穷的非线性问题提供了富有成效的理论工具，处理实际问题时发挥着不可替代的作用，对于这类方程的解也因此成为了研究的热点和难点之一。本项目主要考虑了具有弱奇性的二阶半正微分方程周期正解的存在性和稳定性，并将其推广到具有弱奇性的二阶非自治半正耦合方程组的研究中。我们巧妙的构造格林函数，并利用Leray-Schauder度理论和不动点定理的新技巧和新方法研究弱奇性的二阶半正微分方程周期正解的存在性。利用扭转系数理论构造扭转系数，并通过估计其上下界研究方程及方程组周期正解的稳定性。脉冲现象在自然界中普遍存在，被应用于物理、生物学等诸多领域，本项目还探索性的研究了具有脉冲项影响的奇异耦合方程组周期正解的存在性和稳定性。