时标动态方程边值问题解的分歧结构研究

项目名称： 时标动态方程边值问题解的分歧结构研究

项目编号： No.11301059

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 罗华

作者单位： 东北财经大学

项目金额： 23万元

中文摘要： 线性问题谱理论在非线性问题可解性的研究中起着关键作用. 但时标上已有非线性边值问题的可解性结果并没有明确非线性问题可解性与对应线性问题谱结构间的联系. 本项目从明确这种联系入手, 研究时标非线性边值问题解的全局分歧结构, 分别讨论一般形式的时标非线性二阶加权动态方程边值问题、二阶共振边值问题和四阶边值问题的可解性，获得与相应线性问题谱结构有关的最优的可解性结果. 同时根据研究需要, 发展对应的时标线性问题的特征函数的广义零点分布和正交性等谱结果. 本项目的研究预计可以建立任意2n个解甚至无穷多个解的存在性定理, 同时精确指出每个解在给定区间内的零点个数和变号次数, 推广、统一、发展前人关于常微分方程、差分方程以及时标动态方程边值问题已建立的相应结果, 获得新的可解性定理, 同时丰富时标线性特征值问题的谱理论结果.

中文关键词： 时标；边值问题；解；分歧；特征值

英文摘要： The spectrum theory of the linear problems plays a crucial role in the solvability of the nonlinear problems. But the relationship between the results of solvability of nonlinear boundary value problems on time scales and the spectrum structures of the corresponding linear problems is still unclear.This project is aimed at to study the relationship. The global bifurcation structure of solutions to nonlinear boundary value problems will be discussed, for the generalized form of a nonlinear boundary value problem of second-order dynamic equation with weight function, a nonlinear resonance boundary value problems of second-order dynamic equation and a nonlinear boundary value problem of fourth-order dynamic equation, respectively. The optimal solvability results related to the corresponding linear problems will be obtained. Meantime, as the request of the research, the spectrum results such as the distribution and orthogonal properties of generalized zeros of eigenfunctions of the linear problems on time scales will be developed.The existence theorems for any 2n or infinite solutions will be established in this project. It can pinpoint the numbers of the zeros and the variation of the sign on a given interval of any solutions. The study could generalize, integrate and develope the previous results of boundary value

英文关键词： time scales；boundary value problems；solutions；bifurcation；eigenvalue