二阶线性差分算子的Fucik谱及相关问题研究

In the study of ecology, physics and epidemiology, a large number of nonlinear difference equation models have been developed, which attracted widespread attention. It is well-known that the spectrum of linear operator is one of the most efficient tools to study the nonlinear problems. However, due to the intristic differences between the difference and differential operators, there are many difficulties in the study of difference operator. Many methods which are used to deal with the spectrum of differential operator can not be used to the study of difference operator. Therefore, it is very important to study the spectrum of linear difference operator and the related problems. In this project, by using the matrix theory, polynomial theory, variation method and bifurcation theory, we will study the Fucik spectrum of second-order linear difference operator under the Sturm-Liouville boundary condition and periodic boundary condition respectively. Meanwhile, based on the Fucik spectrum of linear difference operator and by using bifurcation theorem and variation method, we will study the existence, nonexistence and multiplicity of solutions to the nonlinear difference equations boundary value problem with jumping nonlinearity, and attempt to obtain the global structure of the set of solutions.

在生态学、物理学以及流行病学等学科的研究中，产生了大量的非线性差分方程模型，并引起了人们广泛的关注。众所周知，线性算子的谱理论是研究非线性问题最有效的工具之一。但是，由于差分算子和微分算子之间的本质差异，从而导致了许多本质的困难，使得诸多用于微分算子谱理论的研究方法不再适用差分算子谱理论的研究。因此，关于线性差分算子谱理论及其相关问题的研究是具有重要意义的课题。本项目将运用矩阵理论、多项式理论、变分方法以及分歧理论等理论和方法系统地研究由Sturm-Liouville 边界条件和周期边界条件所界定的二阶线性差分算子的Fucik谱。同时，基于线性差分算子的Fucik谱，本项目将运用分歧理论以及变分原理等工具，较为系统地研究具跳跃非线性项的非线性差分方程边值问题解的存在性、不存在性和多解性，并期望获得解集的全局结构。

在生态学、物理学以及流行病学等学科的研究中，产生了大量的非线性差分方程模型，并引起了人们广泛的关注。众所周知，线性算子的谱理论是研究非线性问题最有效的工具之一。但是，由于差分算子和微分算子之间的本质差异，从而导致了许多本质的困难，使得诸多用于微分算子谱理论的研究方法不再适用差分算子谱理论的研究。因此，关于线性差分算子谱理论及其相关问题的研究是具有重要意义的课题。本项目运用矩阵理论、多项式理论、变分方法以及分歧理论等理论和方法系统地对由Sturm-Liouville 边界条件和周期边界条件所界定的二阶线性差分算子的Fucik谱以及边界条件含有谱参数的二阶线性差分算子的谱结构进行了研究。同时，基于所获得的线性差分算子谱结构，本项目运用变分原理、拓扑度理论、不动点指数理论以及分歧理论等工具，对相应的二阶乃至四阶非线性差分方程边值问题解的存在性、不存在性和多解性进行了研究，并获得了解集的全局结构。