带权函数的二阶线性常微分方程多点边值问题的谱及其应用

It can be seen that the study on the solutions set of the boundary value problems of the nonlinear ordinary differential equations is a hot and difficult problem in the field of differential equations, and the study on the problem is closely related to the fact that whether the spectrum of the corresponding linear boundary value problems is completed or not. As we know, there are many important results on the spectrum of the linear second-order two-point boundary value problems, and these results have been widely used in the study on the global structure of the solutions set of the corresponding nonlinear problems. However, the development of the spectral theory of the linear second-order multi-point problems is slow, some previous results depend on the fact that the weight function always equals to 1. In the case of the weight function is not a constant, the study on the spectrum of the linear second-order multi-point is in the initial stage. Therefore, it gives us difficulties to study the global structure of the solutions set of the nonlinear problems. Thus, using the theory of linear operator, the stability theory of ordinary differential equations, and some methods in nonlinear analysis, we'll attempt to study the spectrum of linear second-order multi-point boundary value problems with the weight function and the global structure of the nodal solutions set of the corresponding nonlinear problems. Our results will essentially improve and generalize some previous results in the field, and be of guiding significance to the theory analysis and numerical calculation of some models in control theory and population genetics, etc.

非线性常微分方程边值问题解集分支的全局结构研究是微分方程领域中一个热点和难点问题，而此类问题的研究与相应线性边值问题的谱理论是否完善密切相关。众所周知，二阶线性两点边值问题的谱结构已有许多重要且深刻的结果，并被广泛应用于相应非线性问题解集分支全局结构的研究中。然而，二阶线性多点边值谱理论发展却相对迟缓，仅有的谱结果都是在权函数恒等于1的情形下建立的，当权函数变化时二阶线性多点边问题的谱结构还处于探索阶段，这也使得相应非线性问题解集分支的全局结构研究面临巨大挑战。因此，本项目试图运用线性算子理论、微分方程定性理论和非线性分析中的方法，深入研究带权函数的二阶线性多点边值问题的谱结构，并进一步讨论相应非线性问题结点解分支的全局结构。本项目的研究成果将从本质上改进、丰富和发展该领域的一些已有结果，并对控制理论、人口遗传学等学科中提出的诸多模型的理论分析和数值计算具有重要意义。

本项目主要研究非局部边界条件下带权函数的各类线性特征值问题的谱理论及其相关非线性问题解集连通分支的全局结构。具体工作包括：1、在权函数满足更一般的条件下(权函数只需要是大于等于零的连续函数，并且在其值域区间内有一点大于零即可)，获得了二阶线性微分方程多点边值问题的部分谱结果；2、建立了非局部边值问题适用的全局分歧定理的基础上，研究了相应非线性问题正解解集的全局结构；3、研究了带p-Laplacian算子的线性特征值问题的谱理论、相应的非线性问题的全局分歧定理及结点解集的全局结构；4、运用变分方法获得了一类带强Allee影响的非局部椭圆边值问题的正解的存在性及非存在性结果；5、借助于临界点理论，证明了两类扰动拟椭圆边值问题无穷多解的存在性结果；6、建立了一类微分系统解集结构的通有性结果；7、运用Dancer的单边全局分歧定理，建立了一类半线性算子方程的单边全局分歧理论，并应用其考虑了带不可微非线性项的Sturm-Liouville问题的结点解的存在性和非存在性。