中文摘要
常微分方程边值问题本身具有很强的应用背景，而且又是偏微分方程边值问题研究的基础，有着重要的理论意义和实用价值。当代非线性常微分方程边值问题的研究离不开非线性泛函分析的支持，无论拓扑度理论还是临界点理论，都在近年的研究中得到广泛应用。本课题旨在运用上述两种理论对各类常微分方程边值问题，包括差分方程边值问题和时标上动力系统边值问题进行系统研究，得出有解性和多解性的有效判据，并在理论的应用过程中改进和扩展现有抽象定理的内容和应用领域。三年研究，发表SCI论文90篇，出版著作2本，取得了预期的成果，主要是对带p-Laplace算子的多点边值问题的对称正解和拟对称正解的存在性和迭代方法进行了系统讨论，对多解性作了深入研究，纠正了已有文献中的失误；对二阶和高阶Sturm-Liouville型边值问题的正解存在性给出了有效的判定条件，将这类问题的讨论置于正确的基础之上；研究了时标上动力方程的边值问题，得到了重要的结果；用临界点理论研究带p-Laplace算子常微分方程边值问题，在变分结构的构造等方面有所创新，并得到了理想的结果。
英文摘要
The boundary value problems(BVPs）of ordinary differential equations has, on itself, wide background of applications and serves on the other hand as the basis of the BVPs of partial differential equations. Therefore this research topics is of significance both in theory and in practice. The research of BVPs of nonlinear ordinary differntial equations today can not advance without the support of the theory of nonlinear functional analysis and then both the theory of topological degree and that of critical points are widely applied in the research of BVPs. The aim of our research item is by use of the above mentioned theories to study systematically various types of BVPs of ordinary differential equations, including those of differenc equations and dynamical systems on time scales, to get practical criteria for the existence and multicity of solutions and during the process of research to improve the results and extend the application area of revalent theorems. After three year's study we have pubilished 90 papers in the journals cited by SCI and 2 monographs and then attained our purpose. For example, we systematically discussed the existence and the multicity of symmetrical and pseudo-symmetrical positive solutions and iterative approach for multiple-point BVPs with a p-Laplacian and corrected errors in previous literatures; obtained efficient criteria for the existence of positive solutions to BVPs of 2rd and higher order Sturm-Liouville type and put the discussion on a right basis; researched BVPs of dynamical systems on times scales and got important results; and studied BVPs with a p-Laplacian in applying the theory of critical points and obtained ideal results by our renewing work in the construction of variation structure.
结题摘要
常微分方程边值问题本身具有很强的应用背景，而且又是偏微分方程边值问题研究的基础，有着重要的理论意义和实用价值。当代非线性常微分方程边值问题的研究离不开非线性泛函分析的支持，无论拓扑度理论还是临界点理论，都在近年的研究中得到广泛应用。本课题旨在运用上述两种理论对各类常微分方程边值问题，包括差分方程边值问题和时标上动力系统边值问题进行系统研究，得出有解性和多解性的有效判据，并在理论的应用过程中改进和扩展现有抽象定理的内容和应用领域。三年研究，发表SCI论文90篇，出版著作2本，取得了预期的成果，主要是对带p-Laplace算子的多点边值问题的对称正解和拟对称正解的存在性和迭代方法进行了系统讨论，对多解性作了深入研究，纠正了已有文献中的失误；对二阶和高阶Sturm-Liouville型边值问题的正解存在性给出了有效的判定条件，将这类问题的讨论置于正确的基础之上；研究了时标上动力方程的边值问题，得到了重要的结果；用临界点理论研究带p-Laplace算子常微分方程边值问题，在变分结构的构造等方面有所创新，并得到了理想的结果。