差分方程是现代数学的一个重要分支，有着丰富的实际应用背景。众所周知，在非线性问题的研究中，线性算子的谱扮演着非常重要的角色。因此，本项目将系统地研究二阶非对称线性差分方程变号权特征值问题的谱理论。基于此，本项目将运用分歧理论以及变分原理等工具, 较为系统地研究相应非线性问题解集分支的全局结构以及解的存在性、唯一性和多解性。进一步，将运用全局拓扑扰动方法，研究与非线性微分方程问题相对应的非线性差分方程问题的数值无关解。本项目的理论结果对于计算机科学、经济学、以及种群迁移理论等现代学科中提出的诸多模型的理论分析和数值计算将具有重要意义。

As an important branch of contemporary mathematics，the difference equations have been widely applied in practice. It is well-known that the spectrum of the linear operator plays an important role in the study of the nonlinear problems. Thus, In this project, we attempt to study the spectrum of the second-order linear nonsymmetric difference equations eigenvalue problems. Based on the spectrum, by using the bifurcation theory and the variation principle, etc, we will study the existence, uniquess, multiplicity of solutions to the corresponding nonlinear problems. Furthermore, by using the global topological perbations, we will study the numericall irrelevant solutions of nonlinear difference equation problems corresponding to the nonlinear differential equation problems. The theoretical results of this project will play a significant role in the theoretical analysis and data computation of some models raised in computer science, economics, and metapopulation migration theory, etc.