拟线性广义逆、Bananch流形和非线性方程的分歧分析

The applicant and the first two participants of this project have already assembled a stable strong research group through ten years of collaboration. We study the bifurcation, structure of solution set and other related topics. We have completed two NSFC grants (10671049 and 11071051). This project focuses on the study of C-R bifurcation theorem from a simple eigenvalue and further describes an imperfect bifurcation of small perturbed nonlinear equations. We plan to study the bifurcation theorem from a two-dimensional eigenvalue and obtain Hopf bifurcation theorem and its applications. By using the generalized transversality theorem on Banach manifold, we establish the manifold structure of bifurcation solution set of high dimensional nonlinear equations with parameter and obtain bifurcation theorems on Banach manifold. Furthermore, we describe the perturbation of quasilinear generalized inverse and continuity. For the steady state equation of nonlinear reaction diffusion equation from biology, chemistry and celestial mechanics. We obtain the bifurcation analysis under certain conditions by abstract bifurcation theory. We discuss some properties and applications of the nonlinear stochastic differential equation related to diffusion.

本课题申请人和第1,2参加人经过近十年的合作,已形成稳定的研究团队.关于非线性方程的分歧,解集的结构及其相关课题.已经连续承担完成两次国家基金面上项目(10671049与11071051).本课题研究以从单特征值出发的C-R分歧理论,对小扰动非线性方程的非完美分歧给出进一步刻画. 进一步研究二维特征值出发的分歧定理,以期得到Hopf分歧定理及其应用.应用Banach流形上的广义横截性定理,建立高维带参数非线性方程分歧解集的流形结构,得到Banach流形上的分歧定理.同时,对与其相关的拟线性广义逆的扰动及连续性给出刻画.对于来自生态学,化学动力学,天体力学中的非线性反应扩散方程组及对应的稳态方程,在特定条件下所出现的分歧现象进行分析,既有抽象分歧结果的应用,也有直接进行的分歧分析.研究与扩散相关的非线性随机微分方程的性质及其应用.

本项目运用非线性分析的方法构建了Banach空间中（闭）线性算子的拟线性广义逆的一般扰动理论，其中包括基于广义Banach引理或基于广义Neumann引理的拟线性广义逆的系列扰动定理。由于有界拟线性广义逆中既包含度量广义逆，又包含有界线性广义逆，使得系列已知经典结果成为特例。运用本项目获得的拟线性广义逆存在的充分必要条件，给出Bananch空间中闭线性子空间拓扑可补的充分必要条件，闭子空间拓扑可补在非线性算子的分歧理论中有重要应用。本项目运用获得的全局分歧定理、结合Lery-Schauder度理论、解析分歧理论研究来自生态学、传染病学、自催化理论的反应扩散方程组的解的存在性、非存在性、稳定性、生存性进行研究，得到对应分类的定性分析结果。本项目最后运用非线性随机分析方法，获得随机环境干扰的反应扩散方程组的全局解的存在性、稳定性、及遍历性的刻画,及金融衍生品定价。