测度链上带变号Green函数的非局部问题正解的研究

The theory of time scales, a new analytical theory, was introduced in order to unify continuous and discrete analysis. Boundary value problems have been an important branch of dynamic equations on time scales due to their striking applications to almost all areas of science, engineering and technology. Compared with the classical two-point boundary value problems, generally speaking, the associated differential operators to nonlocal problems are not symmetrical. Up to now, this kind of differential operators has not had integral spectral theory. At the same time, to study positive solutions of nonlocal problems with sign-changing Green's functions on time scales, tremendous difficulties, which are due to the lack of connectivity in the corresponding topological space and the change of sign of Green's functions, need to be overcome. In this item we will study the existence, nonexistence and multiplicity of positive solutions to nonlocal problems with sign-changing Green's functions on time scales by defining some vector fields with crucial properties, controlling these vector fields and combining the method of nonlinear functional analysis.

测度链理论是为了统一连续和离散分析而提出的一种新的分析理论。边值问题由于在科学、工程和技术的几乎所有领域都有着广泛的应用背景而成为测度链上动力方程的一个重要分支。与经典的两点边值问题相比，非局部问题所对应的微分算子一般来说是非对称的，这类微分算子目前还没有完整的谱理论。同时，研究测度链上带变号Green函数的非局部问题的正解需要克服由于相应的拓扑空间缺乏连通性以及Green函数变号所带来的巨大困难。本项目拟定义具有关键性质的向量场并对其进行控制，结合非线性泛函分析的方法对测度链上带变号Green函数的非局部问题正解的存在性、不存在性及多解性进行研究。

测度链理论是为了统一连续和离散分析而提出的一种新的分析理论。边值问题由于在科学、工程和技术的几乎所有领域都有着广泛的应用背景而成为测度链上动力方程的一个重要分支。与经典的两点边值问题相比，非局部问题所对应的微分算子一般来说是非对称的，这类微分算子目前还没有完整的谱理论。同时，研究测度链上带变号Green函数的非局部问题的正解需要克服由于相应的拓扑空间缺乏连通性以及Green函数变号所带来的巨大困难。本项目以不动点指数理论、不动点理论、迭代法、测度链上的微积分理论以及Lyapunov方法等为主要工具不仅在非线性三阶三点边值问题系统、含弯曲项的非线性四阶周期边值问题、带变号Green函数的非线性四阶三点边值问题以及带有积分边界条件的非线性分数阶微分方程非局部问题解和正解的存在性、多解性以及迭代算法等方面获得了一些深刻的结果，而且在测度链上的最优控制问题以及测度链上复杂动态网络的同步问题方面也进行了初步的研究。