發(fā)布時間：2018-04-14 19:33

  本文選題：不連續(xù)神經(jīng)網(wǎng)絡 + 非光滑生物數(shù)學模型　； 參考：《湖南大學》2016年博士論文

【摘要】：近些年來,來源于現(xiàn)實的工程、生物及物理等背景的右端不連續(xù)微分方程及相關問題引起了眾多研究工作者的關注.此外,右端不連續(xù)微分方程所確定的向量場不再是光滑的或全局Lipschitz的,關于右端連續(xù)微分方程的眾多經(jīng)典理論不再適用,致使其理論和研究方法的發(fā)展還遠沒有達到完善的程度.因此,從數(shù)學上對該方程所存在的問題進行深入的探討不僅具有重要的理論意義,而且還具有重大的實際意義.本學位論文綜合利用集值映射理論、微分包含理論、非光滑分析工具以及不等式技巧等數(shù)學理論與方法,特別是右端不連續(xù)泛函微分方程理論及非光滑臨界點理論,并發(fā)展和完善相關的右端不連續(xù)泛函微分方程理論,對幾類具有不連續(xù)激勵函數(shù)的神經(jīng)網(wǎng)絡模型、具有不連續(xù)捕獲項的Lasota-Wazewska模型和Nicholson果蠅模型的動力學性態(tài)進行了定性研究,主要包括平衡點、(概)周期解的存在性、耗散性與(漸近、指數(shù)、有限時間)穩(wěn)定性等問題,并分別研究了一類無界區(qū)域上具有非光滑位勢的Kirchhoff型微分包含問題和一類有界區(qū)域上具有參數(shù)依賴的p(x)-Kirchhoff型微分包含問題,利用非光滑變分原理,分別獲得了所考慮問題新的解的存在性與多重性結果.這些結果既有利于數(shù)學學科的進一步發(fā)展,又為科學和工程應用提供可靠的理論依據(jù)和有效的關鍵技術與方法.全文內容共分為五章,其主要內容如下:在第一章中,回顧了所研究問題的歷史背景、發(fā)展現(xiàn)狀以及最新進展,并對本文的研究工作進行了簡要的陳述,同時也揭示了本論文工作的研究意義和動機.最后,簡單闡述了本論文的研究內容.在第二章中,介紹了本文需要用到的一些基本理論知識,主要涉及集值映射理論、微分包含理論、泛函微分方程、非光滑分析及非光滑變分原理等方面的內容,特別是為研究不連續(xù)微分方程的耗散性,發(fā)展并推廣了一類LaSalle不變原理.在第三章中,利用集值分析理論中的不動點定理,拓撲度理論,并結合非光滑分析技巧、廣義Lyapunov函數(shù)(泛函)方法,以及不等式技巧,分別研究了兩類具有不連續(xù)激勵函數(shù)的神經(jīng)網(wǎng)絡模型及一類憶阻神經(jīng)網(wǎng)絡模型,獲得了相關模型平衡點、周期解的存在性、穩(wěn)定性及耗散性等新結論.同時,本章的部分結果推廣并改進了已有文獻的結論.在第四章中,提出了兩類具有不連續(xù)捕獲項和時滯的Lasota-Wazewska模型和Nicholson果蠅模型,給出了不連續(xù)捕獲項的合理解釋,利用非光滑分析技巧,并發(fā)展了新的分析技巧和方法,得到了所考慮模型(概)周期解的存在性及穩(wěn)定性的全新判據(jù),所建立的(概)周期系統(tǒng)下指數(shù)穩(wěn)定性蘊含其(概)周期解的存在性判據(jù),獲得了這兩類不連續(xù)生物數(shù)學模型(概)周期解的存在性與指數(shù)穩(wěn)定性問題研究的一般方法.在第五章中,首先,研究了一類全空間上Kirchhoff型微分包含問題,克服全空間上Sobolev嵌入緊性和非線性項可微性的缺失所帶來的理論和技術上的困難,利用非光滑版本的山路引理,并結合變分方法,建立了所考察問題解的存在性的新結論.此外,利用非光滑版本的三臨界點定理,并結合變指數(shù)Lebesgue與Sobolev空間理論,研究了一類p(x)-Kirchhoff型微分包含系統(tǒng)在有界區(qū)域上解的存在性問題,建立了所考慮問題解的存在性和多重性等新結果.
[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is not smooth or global Lipschitz, many classical theories on the right end of the continuous differential equation is no longer applicable, resulting in the development of the theory and research methods are still far from perfect.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.

【學位授予單位】：湖南大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O175

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