脉冲发展包含解集的拓扑结构

发布时间：2017-12-27 22:20

本文关键词：脉冲发展包含解集的拓扑结构　出处：《湘潭大学》2016年博士论文　论文类型：学位论文

【摘要】：脉冲微分方程和包含非常适合描述应用科学(如生物学,工程学,经济学,物理学,医学等)领域的系统瞬时突变现象.因此,近年来,脉冲微分方程和包含在建立实际过程的数学模型方面得到广泛了应用.本文主要研究脉冲发展包含解集的拓扑结构,具体包括三个问题:具有一般算子的脉冲发展包含适度解的存在性以及解集的拓扑结构;具有Hille-Yosida算子的脉冲发展包含积分解集的R_δ性质;具有耗散算子的脉冲发展包含C~0-解集的拓扑结构.在第二章,本文引入一些关于函数空间,弱拓扑,半群理论,多值分析以及一些基本定理等预备知识.在第三章,本文研究一类脉冲发展包含适度解集的拓扑结构.首先在第一节,我们给出脉冲微分包含适度解的定义.在第二节中,我们假设线性部分生成的发展算子是紧的.运用弱拓扑方法,我们讨论了脉冲发展包含适度解的存在性,并证明了解集是非空紧R_δ-集.第三节致力于研究线性部分生成的发展算子是非紧的情况.同样运用弱拓扑方法,我们得到了脉冲发展包含适度解的存在性,并证明了解集是非空弱紧R_δ-集.由于我们是在弱拓扑意义下假设非线性项的正则性,因此我们不需要发展算子的紧性条件以及多值非线性项任何关于非紧测度的条件.作为一个简单的应用,我们在本章的最后考虑一个脉冲偏微分包含的例子.在第四章,本文考虑一类具有非稠定闭线性算子的脉冲发展包含积分解集的拓扑结构.脉冲发展包含积分解的定义在第一节中给出.接下来,在半群是紧的和非紧的两种情况下,我们证明在紧区间上脉冲发展包含积分解集是一个紧R_δ-集.利用逆极限方法,我们得到非紧区间上相应的结果.具体的说,第二节致力于半群是紧的情况;第三节则利用非紧测度理论处理半群是非紧的情况.在第五章,本文讨论一类脉冲发展包含C~0-解集的拓扑结构.第一节给出了脉冲微分包含C~0-解的定义.第二节就算子半群是紧的情况,先证明了脉冲发展包含在紧区间上的C~0-解集是非空紧R_δ-集,然后利用逆极限方法,研究非紧区间上C~0-解集的R_δ性质.在第三节,研究半群是等度连续的情况下,脉冲发展包含C~0-解集的R_δ-结构.同样地,先考虑紧区间上C~0-解集的R_δ性质,然后用逆极限方法得到非紧区间上的相应的结果.
[Abstract]:Impulsive differential equations and is very suitable for describing the application of Science (including biology, engineering, economics, physics, medicine, etc.) in the field of instantaneous mutation phenomenon. Therefore, in recent years, impulsive differential equations and establish mathematical model included in the actual process is widely used. This paper mainly studies the development of pulse topology contains a set of solutions, including three problems: the existence of impulsive and topological structure of the solution set appropriate solution development contains general operator; R_ delta properties development include integral solution set pulse with Hille-Yosida operator; with dissipative operator pulse development contains C~0- set topology. In the second chapter, this paper introduces some function space, weak topology, semigroup theory, multi value analysis and some basic theorems of preliminary knowledge. In the third chapter, the development of a class of impulsive contains mild solution set in this paper Topological structure. In the first section, we give the definition of the appropriate pulse solutions to differential inclusions. In section second, we assume that the linear part of the development of operator generated is tight. The weak topology method, we discuss the development of pulse contains the existence of mild solutions, and prove that the solution set is nonempty compact R_ delta set. The third section is committed to the development of the linear part operator generated is non compact. Also the use of weak topological methods, we obtained the pulse development includes the existence of mild solutions, and prove that the solution set is nonempty weakly compact R_ delta set. Since we are assuming the regularity of nonlinearity in the weak topology. Therefore, the compactness condition we do not need the development of operator and multi valued nonlinear term on any measure of non compactness conditions. As a simple application, we have a partial differential pulse contains examples in the final consideration in this chapter. In the fourth chapter In this paper, we consider a class of non dense topology pulse development contains integral solution set of closed linear operators. With development of pulse contains definitions of integral solutions are given in the first section. Then, in the two kinds of semigroups is compact and non compact case, we prove that in the tight interval pulse contains integral development the solution set is a compact R_ delta set. By using the inverse limit method, we obtain the corresponding non compact intervals. Specifically, the second section is devoted to the semigroup is tight; the third section using the measure of noncompactness theory of semigroup is noncompact. In the fifth chapter, this paper discusses a class of impulsive development includes C~0- topology solutions. The first section gives the definition of solutions of impulsive differential inclusions in C~0-. The second section semigroup is tight, to prove that the pulse development included in the compact intervals of C~0- solution set is nonempty compact R_ delta set, and then use the inverse limit method Study on the properties of R_ Delta, non compact intervals C~0- set. In the third section, study of semigroups is equicontinuous case, pulse development includes R_ Delta C~0- solution set structure. Similarly, first consider the R_ delta property of compact interval C~0- set, and then use the method of non inverse limit the tight interval on the corresponding results.
【学位授予单位】：湘潭大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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