非线性差分方程的同宿轨与异宿轨研究

The problems on homoclinic orbits and heteroclinic orbits are of great importance in the theory of difference equations. By using critical point theory, index theory and fixed point theory and combining Nehari manifold method, this project is devoted to studying the existence and multiplicity of homoclinic orbits and heteroclinic orbits of nonlinear difference equations. For difference equations with variational structure, the variational framework on suitable function space is established. By making use of Minimax methods, Morse theory and so on, we set up the theorems of existence and multiplicity of critical points on the space, its subspace or Nehari manifold defined on it, which the variational functional is established. Then these theorems are used to study the problems of homoclinic orbits and heteroclinic orbits. For difference equations without variational structure, the problems of homoclinic orbits and heteroclinic orbits by applying fixed point theory and approximation theory by virtue of operator theory and degree theory are studied. This project will provide some new theories and approaches to study discrete systems and fill the gaps in this field. It will also motivate the development of qualitative theory of difference equations. This project is meaningful in theory and useful in applying.

在差分方程理论中，同宿轨与异宿轨一直是一个重要问题。本项目拟利用临界点理论、指标理论和不动点理论，结合Nehari 流形方法，研究非线性差分方程同宿轨、异宿轨的存在性与多重性。对具变分结构的差分方程，在适当的函数空间建立变分框架，应用Minimax方法、Morse理论等工具，在变分泛函建立的空间、子空间或其上定义的Nehari 流形上，建立变分泛函临界点的存在性和多重性定理，用来研究其同宿轨与异宿轨问题。对不具变分结构的差分方程，以算子理论和度理论为基础，将不动点理论和逼近理论综合运用，研究其同宿轨与异宿轨问题。本项目将为研究离散系统提供新的理论与方法，所得结论填补这一领域的研究空白。这项研究对差分方程定性理论的发展有重要的促进作用，具有重要的理论意义和广泛的应用价值。

本项目利用临界点理论，研究非线性差分方程同宿轨、异宿轨的存在性与多重性。在适当的函数空间建立变分框架，在变分泛函建立的空间建立变分泛函临界点的存在性和多重性定理。利用临界点理论结合周期约化方法，建立了一些具周期系数非线性差分方程同宿轨的存在性与多重性准则。利用临界点理论结合周期约化方法，建立了具周期系数非线性差分方程同宿轨的存在性与多重性存在性准则。利用临界点理论，获得了离散非线性薛定谔方程驻波解（同宿轨）存在性新的结果。应用对称山路引理，获得了离散非线性薛定谔方程一些基态解（同宿轨）多重性结果的充分条件。考虑一类二阶非线性差分方程，利用临界点理论，获得了两条异宿轨存在性新的结果。