非线性差分方程的最小周期解与边值问题研究

Periodic phenomena and boundary value problems occur in various fields of natural science. Periodic solutions especially on minimal periodic solutions and boundary value problems attract attentions of many famous mathematicians all around the world. Using critical point theory, this project is devoted to studying the minimal periodic solutions and boundary value problems of nonlinear difference equations. The variational framework on suitable function space for difference equations is established. By applying the duality variational methods, perturbation argument, dual least action principle, minimax methods, Morse theory and geometrical index theory, we set up the theorems of existence and multiplicity of critical points on the space which the variational functional is established. Then these theorems are used to study the problems of minimal periodic solutions. The existence and mutiplicity of positive solutions, negative solutions and sign-changing solutions to boundary value problems for difference equations are obtained via computing precisely the corresponding critical groups, by using Morse theory, cut-off technique, Mountain Pass Lemma, Linking Theorem, degree theory and matrix analysis, etc. What's more, the number of solutions is more precisely estimated. Some new results are obtained and this project offers an attempt to use critical point theory to deal with the minimal periodic solutions and boundary value problems of difference equations. Difference equations have widely occurred in probability theory, matrix theory, queuing theory, number theory, psychology and sociology, etc. This project is of important significance both theoretically and practically.

周期现象与边值问题普遍存在于自然科学的各个研究领域，周期解特别是最小周期解与边值问题一直是广大学者关注的问题。本项目拟应用临界点理论研究非线性差分方程最小周期解与边值问题。在适当的函数空间建立变分框架，利用对偶变分法、扰动技巧、对偶最小作用原理、极小极大方法、Morse理论和几何指标理论，在变分泛函建立的空间上，建立变分泛函临界点的存在性定理和多重性定理，用来研究差分方程最小周期解。运用Morse理论、截断技巧、山路引理、环绕定理、度理论与矩阵分析等，精确计算差分方程中相应的临界群，研究其各种边值问题正解、负解与变号解的存在性与多重性，并对解的个数作较精确的估计。本项目将获得一些新的研究成果，对临界点理论在差分方程最小周期解与边值问题中的应用作出新的尝试。由于差分方程广泛出现在概率论、矩阵论、排队论、数论、心理学与社会学等领域中，本项目具有重要的理论意义和广泛的应用价值。

周期现象与边值问题普遍存在于自然科学的各个研究领域，周期解特别是最小周期解与边值问题一直是广大学者关注的问题。本课题利用变分法、临界点理论研究非线性差分方程最小周期解与边值问题。在适当的函数空间建立变分框架，在变分泛函建立的空间上，建立变分泛函临界点的存在性定理和多重性定理。利用临界点理论，获得了二阶、四阶、2n阶p-Laplacian非线性差分方程以及具有超前和滞后的高维差分系统边值问题解的存在性定理、多重性、唯一性、以及不存在性新的结果。利用变分法以及临界点理论，获得了四阶非线性差分方程最小周期解的存在性定理和多重性新的结果。