右端不连续微分方程理论及相关问题研究

发布时间：2018-04-14 19:33

本文选题：不连续神经网络 + 非光滑生物数学模型　；参考：《湖南大学》2016年博士论文

【摘要】：近些年来,来源于现实的工程、生物及物理等背景的右端不连续微分方程及相关问题引起了众多研究工作者的关注.此外,右端不连续微分方程所确定的向量场不再是光滑的或全局Lipschitz的,关于右端连续微分方程的众多经典理论不再适用,致使其理论和研究方法的发展还远没有达到完善的程度.因此,从数学上对该方程所存在的问题进行深入的探讨不仅具有重要的理论意义,而且还具有重大的实际意义.本学位论文综合利用集值映射理论、微分包含理论、非光滑分析工具以及不等式技巧等数学理论与方法,特别是右端不连续泛函微分方程理论及非光滑临界点理论,并发展和完善相关的右端不连续泛函微分方程理论,对几类具有不连续激励函数的神经网络模型、具有不连续捕获项的Lasota-Wazewska模型和Nicholson果蝇模型的动力学性态进行了定性研究,主要包括平衡点、(概)周期解的存在性、耗散性与(渐近、指数、有限时间)稳定性等问题,并分别研究了一类无界区域上具有非光滑位势的Kirchhoff型微分包含问题和一类有界区域上具有参数依赖的p(x)-Kirchhoff型微分包含问题,利用非光滑变分原理,分别获得了所考虑问题新的解的存在性与多重性结果.这些结果既有利于数学学科的进一步发展,又为科学和工程应用提供可靠的理论依据和有效的关键技术与方法.全文内容共分为五章,其主要内容如下:在第一章中,回顾了所研究问题的历史背景、发展现状以及最新进展,并对本文的研究工作进行了简要的陈述,同时也揭示了本论文工作的研究意义和动机.最后,简单阐述了本论文的研究内容.在第二章中,介绍了本文需要用到的一些基本理论知识,主要涉及集值映射理论、微分包含理论、泛函微分方程、非光滑分析及非光滑变分原理等方面的内容,特别是为研究不连续微分方程的耗散性,发展并推广了一类LaSalle不变原理.在第三章中,利用集值分析理论中的不动点定理,拓扑度理论,并结合非光滑分析技巧、广义Lyapunov函数(泛函)方法,以及不等式技巧,分别研究了两类具有不连续激励函数的神经网络模型及一类忆阻神经网络模型,获得了相关模型平衡点、周期解的存在性、稳定性及耗散性等新结论.同时,本章的部分结果推广并改进了已有文献的结论.在第四章中,提出了两类具有不连续捕获项和时滞的Lasota-Wazewska模型和Nicholson果蝇模型,给出了不连续捕获项的合理解释,利用非光滑分析技巧,并发展了新的分析技巧和方法,得到了所考虑模型(概)周期解的存在性及稳定性的全新判据,所建立的(概)周期系统下指数稳定性蕴含其(概)周期解的存在性判据,获得了这两类不连续生物数学模型(概)周期解的存在性与指数稳定性问题研究的一般方法.在第五章中,首先,研究了一类全空间上Kirchhoff型微分包含问题,克服全空间上Sobolev嵌入紧性和非线性项可微性的缺失所带来的理论和技术上的困难,利用非光滑版本的山路引理,并结合变分方法,建立了所考察问题解的存在性的新结论.此外,利用非光滑版本的三临界点定理,并结合变指数Lebesgue与Sobolev空间理论,研究了一类p(x)-Kirchhoff型微分包含系统在有界区域上解的存在性问题,建立了所考虑问题解的存在性和多重性等新结果.
[Abstract]:In recent years, from the practical engineering, biological and physical background of the discontinuous differential equations and related problems have attracted many attentions. In addition, the vector field differential equations with discontinuous right-hand side determined is not smooth or global Lipschitz, many classical theories on the right end of the continuous differential equation is no longer applicable, resulting in the development of the theory and research methods are still far from perfect. Therefore, not only has important theoretical significance for in-depth discussion of the equation in mathematics problems, but also has great practical significance. The theory of comprehensive utilization of set-valued mapping in this thesis, the theory of differential inclusions, nonsmooth analysis theory and method of mathematical tools and techniques of inequalities, especially the discontinuous functional differential equation theory and nonsmooth critical point theory, and the development and perfect Close the right end of the discontinuous functional differential equation theory, neural network models with discontinuous activation functions of several types, with dynamic discontinuity capturing Lasota-Wazewska model and Nicholson model of the Drosophila of qualitative research, including the balance point, (almost) the existence of periodic solutions, and dissipative (Co. asymptotic stability, index) problem, and investigated a class of unbounded domains with Nonsmooth potential Kirchhoff problems and a class of differential inclusions with parameter dependent P bounded region (x) of -Kirchhoff type differential inclusions, using nonsmooth variational principle are obtained by considering the problem of existence and multiplicity results of solutions. These new results not only conducive to the further development of mathematics, but also provide a reliable theoretical basis and effective key technologies and methods for science and engineering applications. The full text Content is divided into five chapters, the main contents are as follows: in the first chapter, reviews the research questions the historical background, development status and the latest progress, and the research work of this paper makes a brief statement, but also reveals the significance and motivation. Finally, briefly discusses the the research content of this thesis. In the second chapter, this paper introduces the need to use some of the basic theoretical knowledge, mainly involving set-valued mapping theory, the theory of differential inclusions, functional differential equations, nonsmooth analysis and nonsmooth variational principle and other aspects, especially for the study of discontinuous dissipative differential equations, development and the promotion of a kind of LaSalle invariant principle. In the third chapter, by using set-valued analysis in the theory of fixed point theorem, topological degree theory, and combining with nonsmooth analysis techniques, generalized Lyapunov function (functional) method and inequality technique, divided Don't study two kinds of neural network models with discontinuous activation functions and a class of memristive neural network model has obtained the equilibrium points, the existence of periodic solutions, stability and dissipation of the new conclusion. At the same time, part of the results in this chapter improve and generalize the related results in the literatures. In the fourth chapter and put forward two kinds of Lasota-Wazewska model and Nicholson model of Drosophila discontinuity capturing and time-delay, is given a reasonable explanation of the continuous acquisition, using nonsmooth analysis techniques, and analysis of the development of new skills and methods, has been considered model (almost) a new criterion for the existence and stability of periodic solution. And the (almost) cycle system contains the stability index (almost) existence criterion of periodic solutions obtained, these two kinds of discontinuous biological mathematical model (almost) of number and refers to the stability problem of Periodic Solutions The general method. In the fifth chapter, first of all, the whole space is studied for a class of Kirchhoff differential inclusion problems, overcome brought Sobolev embedding compactness and nonlinear differentiable loss of the whole space theory and technical difficulties, the use of non smooth version of the mountain pass lemma, and combining with the variational method, the establishment of the investigation of new existence results of solutions of the problems. In addition, the three critical point theorem of non smooth version, combined with the variable index Lebesgue and Sobolev space theory, study a class of P (x) system in the existence of solutions of bounded domain -Kirchhoff type differential inclusions are established considering the existence of and the multiplicity of new solutions.

【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

【相似文献】