在物理学、工程学以及天文学等学科中,产生了大量边界条件含有谱参数的差分方程模型,并引起了人们广泛的关注。但是,由于边界条件和方程中均含有谱参数以及微分方程与差分方程之间的内在差异,导致在边界条件含有谱参数的线性差分算子谱理论的研究中出现了许多本质的困难。这严重制约了该类线性差分算子的谱理论及相应非线性问题解的存在性的研究。因此,对边界条件含有谱参数的线性差分算子的谱及其非线性应用的研究是差分方程研究领域具有重要理论意义和应用价值的研究课题。基于此,本项目将系统地研究如下问题:首先,研究边界条件含有谱参数的二阶线性差分算子的谱;其次,研究边界条件含有谱参数(对)的二阶线性多参数差分系统的谱;再次,基于所获谱结构,研究相应的二阶非线性差分方程或差分系统正解和变号解的存在性。本项目的预期成果不仅是对线性差分算子谱理论的丰富和发展,而且为相关应用学科领域的研究奠定坚实的数学理论基础。

Lots of models of difference equation with eigenparameter-dependent boundary conditions have been constructed in academic fields such as physics,engineering and astronomy,and have attracted extensive attention.However,since both the boundary condition and the equation contain the eigenparameters and the inherent differences between the difference equations and the differential equations,lots of intrinsic difficulties appear in the study of the spectral theory of linear difference operator with eigenparameter-dependent boundary conditions.Furthermore,these difficulties severely restrict the study of the spectral theory of this kind of linear difference operator and the existence of solutions to the corresponding nonlinear problems.Therefore,the study of eigenvalue problems of difference equations with eigenparameter-dependent boundary conditions has theoretical significance and practical values.Based on these,this project will systematically study the following problems:first,the spectrum of linear difference operator with eigenparameter-dependent boundary conditions;second,the spectrum of linear multiparameter difference systems with eigenparameter-dependent boundary conditions,and third, the existence of positive solutions and sign-changing solutions of nonlinear difference equations and difference systems with eigenparameter-dependent boundary conditions.The expected results of this project will not only enrich and develop the spectral theory of the linear difference operator,but also lay the foundation of the mathematical theory for the study of the related applied disciplines.