本文關鍵詞： 耗散系統(tǒng) 旋轉(zhuǎn)周期解 脈沖條件 微分包含 Horn不動點定理 非凸變分問題　出處：《吉林大學》2016年博士論文　論文類型：學位論文

【摘要】：在對各類自然現(xiàn)象進行觀察的過程中,人們發(fā)現(xiàn)大千世界中的許多現(xiàn)象和過程由于某種原因,其狀態(tài)或行為在某些時間和某種環(huán)境下會發(fā)生突然的改變或明顯的變化,而通常的微分方程在這樣的情況下已經(jīng)很難精確描述這種突變的規(guī)律,因此人們開始尋找能夠更準確的描述這種突變現(xiàn)象的數(shù)學模型.脈沖微分方程的相關理論就在這一背景下應運而生.脈沖微分方程的系統(tǒng)理論起源于20世紀60年代V. D. Mil'man和A. D. Myshkis[1]合作發(fā)表了關于脈沖外力作用下物體運動穩(wěn)定性的論文,這是脈沖微分方程領域最早的有影響力的成果.這一類方程的最大特點就是能夠反映出系統(tǒng)在受到瞬時突變影響時狀態(tài)的變化,因此在人口學、物理學、生物學以及控制理論等許多學科中都有著廣泛的應用背景.同時,在描述某些狀態(tài)不確定或者存在某種多值性的系統(tǒng)時,微分包含是比微分方程更加合適的數(shù)學工具.本篇博士論文便是以帶有脈沖條件的微分方程以及微分包含為對象,討論了當這類系統(tǒng)具有耗散性時,其周期解以及旋轉(zhuǎn)周期解的存在性問題.本篇博士論文的結(jié)構(gòu)如下.在第一章中,我們簡要介紹了與旋轉(zhuǎn)周期性和脈沖現(xiàn)象有關的一些基本定義和重要結(jié)果.在第二章中,我們以耗散脈沖微分方程以及耗散泛函脈沖微分方程為研究對象,即系統(tǒng)以及系統(tǒng)在2.1節(jié)中,我們首先證明了一個耗散的脈沖微分方程的周期解的存在性,也就是下面的定理.定理0.0.1設(0.0.1)是周期脈沖系統(tǒng),f和{Ii}i∈Z1關于x滿足局部Lipschitz條件,{τi(x)}和{Ii(x)}滿足條件i)和ii).如果系統(tǒng)(0.0.1)是耗散的,則(0.0.1)有一個T-周期解.隨后,我們考慮一種更加一般化的周期解,也就是旋轉(zhuǎn)周期解,并在2.2節(jié)中證明了耗散的脈沖微分方程的旋轉(zhuǎn)周期解的存在性定理.定理0.0.2設(0.0.1)是一個(Q,T)-周期脈沖系統(tǒng),f和{Ii}i∈Z1關于x滿足局部Lipschitz條件,且{τi(x)}和{Ii(x)}滿足條件i)和ii).如果系統(tǒng)(0.0.1)是Q-耗散的,則該系統(tǒng)一定有一個(Q,T)-旋轉(zhuǎn)周期解.在第二章的最后,我們討論了耗散的泛函脈沖微分方程,并證明了這類系統(tǒng)旋轉(zhuǎn)周期解的存在性定理.定理0.0.3如果系統(tǒng)(0.0.2)是(Q,T)-耗散的,則其存在(Q,T)-旋轉(zhuǎn)周期解.正如前文所述,在討論某些特定問題時,因為研究對象的狀態(tài)具有某種不確定性或者是多值性,在很多情況下微分包含是比微分方程更加合適的數(shù)學工具.因此在第三章中我們討論了具有耗散性和脈沖條件的微分包含以及泛函微分包含的旋轉(zhuǎn)周期解的存在性.即系統(tǒng)以及系統(tǒng)在3.2節(jié)中,我們證明了一個耗散的脈沖微分包含的旋轉(zhuǎn)周期解的存在性定理.定理0.0.4假設f(t,x):R×Rn→K(Rn)是上半連續(xù)的,且是(Q,T)-旋轉(zhuǎn)周期的,即對所有的(t,x)∈R×Rn,有其中Q∈GL(n),T0是一個常數(shù),并且一個脈沖條件下的微分包含的解是一致M最終有界的,則微分包含(0.0.7)有(Q,T)-旋轉(zhuǎn)周期解.隨后,類似于第二章,我們討論了同樣條件下的泛函微分包含,并證明了旋轉(zhuǎn)周期解的存在性.定理0.0.5假設F:R×Cτ([一r,0])→Comp(Rn)滿足如下條件.(H1) F是下半連續(xù)的;(H2)對任意t∈R,φ∈Cτ([-r,0])滿足||φ||≤M,存在L(M)使得d(F(t,φ),0)≤L(M);(H3)對任意的t∈R,φ∈Cτ([-r^,0]),都有F(t+T,φ)=QF(t,Q-1φ),其中Q∈GL(Rn),T0為常數(shù).令Do (?) D1為Rn中的有界子集,Do為閉集,D1為凸開集.那么,如果泛函微分包含(0.0.5)-(0.0.6)的解是D1-有界的并且是D1-Do局部旋轉(zhuǎn)耗散的,則泛函微分包含(0.0.5)-(0.0.6)在Do中有(Q,T)-旋轉(zhuǎn)周期解.我們在最后一章考慮如下具有三階Lagrange函數(shù)的一維非凸變分問題其中f：[0,1]×R4→R1是連續(xù)函數(shù),x∈W2,1(0,1),x(0)=x0,z(1)= x1,x'''于(0,1)內(nèi)幾乎處處存在.該問題的研究對于非凸變分學的發(fā)展和非凸Hamilton系統(tǒng)的Mather理論均有積極意義.在適當條件下,我們利用積分-極大極小方法,給出了上述變分問題最優(yōu)解存在的一個充分條件.事實上,該結(jié)論可推廣至更為一般的高階非凸變分問題.
[Abstract]:In all kinds of natural phenomena in the process, people found that many of the universe of 1,000,000,000 universes phenomena and processes for some reason, the state or behavior will be a sudden change or changes in certain times and environment, and the differential equations are usually in such cases is difficult to accurately describe the mutation regularity therefore, people began to search for mathematical model to more accurately describe the mutation phenomenon. The theory of impulsive differential equation is in this context. The origin of system theory of impulsive differential equations in 1960s V. D. Mil'man and A. D. Myshkis[1] published on the pulse motion stability under external force, this is the result of impulsive differential equation the earliest influential. The most important feature of this kind of equation is able to reflect the system is instantaneous State change mutation effect, therefore in demography, physics, biology, control theory and so many disciplines have a wide application background. At the same time, in the description of some uncertain state or the existence of a multi value system, differential inclusions is a mathematical tool is more appropriate than the differential equations. This dissertation will with the pulse condition of differential equation and differential inclusions are discussed as the object, when this kind of system is dissipative, the rotation of the periodic solution and the existence of periodic solutions. This dissertation is structured as follows. In the first chapter, we briefly introduce some basic definitions and important results related to rotation periodicity and pulse phenomena. In the second chapter, we take the dissipation of impulsive differential equations and impulsive differential equations with dissipation function as the research object, the system and the system in Section 2.1, we first 璇佹槑浜嗕竴涓，

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