本文選題：指數(shù)二分性 + 指數(shù)三分性　； 參考：《吉林大學(xué)》2016年博士論文

【摘要】：微分方程的指數(shù)二分的概念最早是由Lyapunov和Poincare在19世紀(jì)末提出的,并在隨后的時間里迅速成為了微分方程領(lǐng)域中的重要研究對象之一.1930年,Perron ([24])發(fā)展了線性微分方程的指數(shù)二分理論,并以其為工具研究了線性系統(tǒng)的條件穩(wěn)定性問題.從那時起,指數(shù)二分的理論便在微分方程領(lǐng)域被數(shù)學(xué)家們廣泛應(yīng)用,其中的部分成果可以參見[12,13,25]及其相關(guān)的文獻(xiàn).在1974年,Sacker和Sell ([28])共同提出了指數(shù)三分的概念并建立了相關(guān)的基本理論.Elaydi和Hajek ([17])隨后研究了微分系統(tǒng)的指數(shù)三分性.在之后的時間里,指數(shù)三分概念作為指數(shù)二分性質(zhì)的推廣,在動力系統(tǒng)相關(guān)領(lǐng)域的研究中同樣發(fā)揮了重要的作用.指數(shù)二分性和指數(shù)三分性是定性理論中非常重要的漸近性.所以研究指數(shù)二分性和指數(shù)三分性是十分必要的.本文給出了微分方程的指數(shù)二分性和指數(shù)三分性及仿射周期解的深入研究.為了使得本文更加獨立,我們在第一章中首先介紹了一些已有的工作,然后給出了要用到的仿射周期解的定義和一些性質(zhì),接著回顧了微分方程的指數(shù)二分性和指數(shù)三分性的概念,最后指出在本文中.我們以仿射周期系統(tǒng)為研究對象,討論指數(shù)二分以及指數(shù)三分條件下仿射周期系統(tǒng)的仿射周期解以及偽仿射周期解的存在性問題.在第二章中,我們討論了具有指數(shù)二分性的一階仿射周期系統(tǒng).首先,考慮了一階線性非齊次方程x'=A(t)x+f(t),其中連續(xù)有界函數(shù)A(t):R1→Rn×Rn和f(t):R1→Rn滿足(Q,T)-仿射周期性,若相應(yīng)的齊次方程x'=A(t)x滿足指數(shù)二分性,我們有如下的定理：定理1如果線性方程x'=A(t)x對于投影P是指數(shù)二分的,A(t)和f(t)滿足(Q,T)-仿射周期性.那么非齊次線性微分方程x'=A(t)x+f(t)存在一個(Q,T)-仿射周期解.其次,考慮了一階半線性微分方程x'=A(t)x+g(t,x(t)),其中g(shù):R1×Rn→Rn連續(xù),A(t)和g(t,x)是(Q,T)-仿射周期的,若相應(yīng)的齊次方程x'=A(t)x滿足指數(shù)二分性,我們有定理2線性微分方程x'=A(t)x對于投影映射P和正常數(shù)K,L,α,β是指數(shù)二分的.同時,假設(shè)A(t),g(t,x)是(Q,T)-仿射周期的,Q∈GL(n)如果g(t,x)是有界函數(shù)滿足利普希茨條件,那么方程x'=A(t)x+g(t,x(t))存在唯一的(Q,T)-仿射周期解.事實上,定理2中關(guān)于g(t,x(t))的利普希茨條件可以由線性增長條件取代,具體定理如下：定理3如果線性微分方程x'=A(t)x對于投影映射P和常數(shù)K,L,α,β0是指數(shù)二分的.同時,假設(shè)A(t),g(t,x)是(Q,T)-仿射周期的,其中Q∈O(n),g(t,x)滿足條件(C1),那么方程x'=A(t)x+g(t,x(t))存在(Q,T)-仿射周期解.在第三章中,我們討論了具有指數(shù)二分性的高階仿射周期系統(tǒng).首先,考慮了n-維二階線性非齊次仿射周期系統(tǒng)x"+p(t)x'+q(t)x= e(t),其中p(t),q(t):R1→Rn×n,e(t):R1→Rn是連續(xù)的、(Q,T)-仿射周期的,給出如下結(jié)果：定理4假設(shè)p(t),(qt)和e(t)是連續(xù)的(Q,T)-仿射周期函數(shù),并且對于所有的t∈R1,F(t),G(t)是有界的.如果p(T)和q(t)滿足下列條件之一：1).對于所有的t∈R1,p(t)和q(t)是正定或負(fù)定的；2).對于所有的t∈R1,q(t)是負(fù)定的,那么,對于所有的t∈R1,F(t)有k個實部小于等于-α(α0)和2n-k個實部大于等于β(β0)的特征值.進(jìn)一步,假設(shè)(?)0εmin(α,β)存在δ=δ(α+β,ε)0使得：如果存在h0滿足對于所有的|t2-t1|≤h,總有|F(t2)-F(t1)|≤δ,則方程x"+p(t)x'+q(t)x=e(t)存在(Q,T)-仿射周期解.其次,考慮了m階線性非齊次系統(tǒng)x(m)=a(t)x+e(t),其中a(t):R1→Rn×n,e(t):R1→Rn連續(xù),滿足(Q,T)-仿射周期條件.具體地,我們給出了以下定理：定理5假設(shè)a(t)和e(t)是連續(xù)的(Q,T)-仿射周期函數(shù),并且對于所有的t∈R1:A(t),G(t)是有界的.如果1).當(dāng)m=4k,k∈Z時,對于所有的t∈R1:a(t)是負(fù)定的;2).當(dāng)m=4k+2,k∈Z,對于所有的t∈R1,a(t)是正定的;3).當(dāng)m=4k+1或4k+3,k∈Z,對于所有的t∈R1,a(t)是正定或負(fù)定的,那么,對于所有的t∈R1,A(t)有k個實部小于等于-α(α0)和mn-k個實部大于等于β(β0)的特征值.進(jìn)一步,假設(shè)(?)0εmin(α,β),存在δ=δ(α+β,ε)0使得：如果存在h0滿足對于所有的|t2-t1|≤h,總有|A(t2)-A(t1)|≤δ,則方程x(m)=a(t)x+e(t)存在(Q,T)-仿射周期解.對于指數(shù)三分性,我們在第四章中討論了具有指數(shù)三分性的仿射周期系統(tǒng).首先,考慮了半線性微分方程x'=A(t)x+g(t,x(t)),其中g(shù):R1×Rn→Rn為連續(xù)函數(shù),A(t)和g(t,x)為(Q,T)-仿射周期函數(shù),其對應(yīng)的齊次線性微分方程為x'=A(t)x.定理6如果方程x'=A(t)x對于投影P1,P2以及常數(shù)K,a是指數(shù)三分的.同時,A(t),g(t,x)是(Q,T)-仿射周期函數(shù),g(t,x)是有界函數(shù)并且對任意的t,x,y,滿足|g(t,x)-g(t,y)|≤N|x-y|,其中Q∈GL(n),N0是一個常數(shù)且使得下式成立則方程x'=A(t)x+g(t,x(t))存在唯一的(Q,T)-仿射周期解.接著,我們給出了偽仿射周期解的定義.對于偽(Q,T)-仿射周期解,我們同樣可以證明下面的存在性定理.定理7對于系統(tǒng)x'=A(t)x+g(t,x(t)),如果A(t)是(Q,T)-仿射周期的,g(t,x)是一個可以分解為g(t,x)=g1(t,x)+g2(t,x)的偽仿射周期函數(shù),其中Q∈GL(n),T0為常數(shù),g1(t,x)∈CT, g2(t,x)∈C0同時,對于Rn的任意有界子集,函數(shù)g(t,x)和g1(t,x)均關(guān)于t∈R1一致連續(xù).函數(shù)g(t,x)滿足|g(t,x)-g(t,y)|≤N|x-y|,(?)t,x,y,其中N0為常數(shù).如果系統(tǒng)x'=A(t)x+g(t,x(t))對應(yīng)的齊次線性方程x'=A(t)x是指數(shù)三分的,并且指數(shù)三分條件中的投影Pl,P2以及常數(shù)K,α滿足適當(dāng)?shù)臈l件,那么系統(tǒng)x'=A(t)x+g(t,x(t))一定存在偽(Q,T)-仿射周期解,并且這個解是唯一的.對于指數(shù)三分性,我們還有如下的推論：推論1如果方程x'=A(t)x+g(t,x(t))對于投影P1,P2以及常數(shù)K,α是指數(shù)三分的,同時,A(t),g(t,x)是(Q,T)-仿射周期函數(shù),g(t,x)對于任意的t∈R1均關(guān)于x一致連續(xù),并且滿足|g(t,x)|≤a|x|+b,(?)t,x,其中Q∈O(n),a,b為大于0的常數(shù),并使得2Kα/α1成立,那么方程x'=A(t)x+g(t,x(t))存在唯一的(Q,T)-仿射周期解.推論2對于系統(tǒng)x'=A(t)x+g(t,x(t)),假設(shè)A(t)是(Q,T)-仿射周期的,g(t,x)是一個可以分解為g(t,x)=g1(t,x)+g2(t,x)的偽仿射周期函數(shù),其中Q∈O(n)(n),T0為常數(shù)g1(t,x)∈CT, g2(t,x)∈C0同時,g(t,x)對于任意的t∈R1均關(guān)于x一致連續(xù),并且滿足|g(t,x)|≤a|x|+b,(?)t,x,其中a,b為大于0的常數(shù).如果系統(tǒng)x'=A(t)x+g(t,x(t))對應(yīng)的齊次線性方程(1.4.13)是指數(shù)三分的,并且指數(shù)三分條件中的投影P1,P2以及常數(shù)K,α滿足條件2Kα/α1,那么系統(tǒng)x'=A(t)x+g(t,x(t))一定存在偽(Q,T)-仿射周期解,并且這個解是唯一的.這些就是本論文的全部內(nèi)容.
[Abstract]:The concept of exponential two points of differential equations was first proposed by Lyapunov and Poincare at the end of nineteenth Century. In the subsequent time, the concept of exponents of differential equations was rapidly becoming one of the most important research objects in the field of differential equations. Perron ([24]) developed the exponential two theory of linear differential equations and studied the conditions of linear systems with its tools. Stability problems. Since then, the theory of exponential two points has been widely used by mathematicians in the field of differential equations. Some of the results can be seen in [12,13,25] and related literature. In 1974, Sacker and Sell ([28]) jointly proposed the concept of index three points and established the relevant basic theories,.Elaydi and Hajek ([17]). The exponent three division of the differential system is investigated. In the following time, the exponent three concept is extended as an exponent of the two nature of the index. It also plays an important role in the research on the related fields of the dynamic system. The index two and the exponent three are very important in the qualitative theory. Therefore, the index two and the index three points are studied. In order to make this paper more independent, we first introduce some existing work in the first chapter, and then give the definition and some properties of the affine periodic solution to be used in the first chapter, and then review the differential square in order to make this paper more independent. In this paper, we take an affine periodic system as the research object, and discuss the existence of the affine periodic solution and the pseudo affine periodic solution of the affine periodic system under the index two points and the index three sub conditions. In the second chapter, we discuss an exponential two partition. First, we consider the first order linear nonhomogeneous equation x'=A (T) x+f (T), in which the continuous bounded function A (T): R1, Rn * Rn and f (T): R1 to satisfy the affine periodicity, if the corresponding homogeneous equation satisfies the exponent two, we have the following theorem: theorem 1 if the linear equation is the exponent of the projection Two points, A (T) and f (T) satisfy (Q, T) - affine periodicity. Then the non homogeneous linear differential equation x'=A (T) x+f (T) exists a (Q, T) - affine periodic solution. Secondly, the first order semilinear differential equation is considered to be an affine periodic, if the corresponding homogeneous equation satisfies the index. Two, we have theorem 2 linear differential equation x'=A (T) x for projection mapping P and normal numbers K, L, alpha, and beta are exponent two. Meanwhile, suppose A (T), G (T, x) is a bounded function satisfying the condition of the Li, then the equation exists only the affine periodic solution. In fact, G (T, X (T)) in theorem 2 can be replaced by linear growth conditions. The specific theorem is as follows: Theorem 3 if the linear differential equation x'=A (T) x is an exponent of the projection mapping P and constant K, L, alpha, and beta 0 are exponentially two. A (T) x+g (T, X (T)) exists (Q, T) - affine periodic solution. In the third chapter, we discuss the high order affine periodic system with exponential two division. Firstly, the n- dimension two order Linear Nonhomogeneous affine periodic system X "+p T) is considered. Theorem 4 assumes that P (T), (QT) and E (T) are continuous (Q, T) - affine periodic functions and are bounded for all t R1, F (T) and G (1). The eigenvalues of K solid parts less than equal to - alpha (alpha 0) and 2n-k real parts are greater than that of beta (beta 0). Further, it is assumed that (?) 0 e min (alpha, beta) exists delta = Delta (alpha + beta, epsilon) 0 so that if H0 satisfies all |t2-t1| < h, there is always |F (T2) -F (T1) < < delta >, then x "+p" exists the affine periodic solution. Secondly, considering the linearity of the order linear solution. The non homogeneous system X (m) =a (T) x+e (T), a (T): R1, Rn x n, e, satisfies the affine periodic condition. Specifically, we give the following theorems: Theorem 5 hypothesis is continuous and affine periodic function, and for all, if 1). R1:a (T) is negative definite; 2). When m=4k+2, K Z, for all t R1, a (T) is positive definite; 3). When m=4k+1 or 4k+3, it is positive definite or negative. The existence of delta = Delta (alpha + beta, epsilon) 0 makes: if the existence of H0 satisfies all |t2-t1| < h, there is always |A (T2) -A (T1) < < delta >, then the equation x (m) =a (T) exists the affine periodic solution. For the exponent, we discuss the affine periodic system with the number of points in the fourth chapter. First, consider the semilinear differential equation. X (T)), in which g:R1 x Rn - Rn is a continuous function, A (T) and G (T, x) are (Q, T) - affine periodic functions. |g (T, x) -g (T, y) less than N|x-y|, where Q GL (n), N0 is a constant and makes the next equation a unique periodic solution. Then, we give the definition of the pseudo affine periodic solution. For the pseudo (pseudo), affine periodic solution, we can also prove the existence theorem. Theorem 7 is a system for the system. X'=A (T) x+g (T, X (T)), if A (T) is (Q, T) - affine cycle, G (T,) is a pseudo - affine periodic function that can be decomposed. (T, y) < N|x-y|, (?) t, x, y, where N0 is constant. If the system x'=A (T) x+g (T, x) is an exponent of the exponent, and a constant, alpha satisfies the proper condition, then the system has a pseudo periodic solution, and this solution is unique. We also have the following inference: deductions 1 if the equation x'=A (T) x+g (T, X (T)) is for the projection of P1, P2 and constant K, alpha is exponentially three, while A (T) is an affine periodic function. The equation x'=A (T) x+g (T, X (T)) exists the only (Q, T) - affine periodic solution. The inference 2 for the system x'=A (T) x+g (T) is a pseudo - affine periodic function that can be decomposed to 2K. G2 (T, x) C0 at the same time, G (T, x) for any t R1 are uniformly continuous on X, and satisfy |g (T,) to be a constant greater than 0. Then the system x'=A (T) x+g (T, X (T)) must have pseudo (Q, T) - affine periodic solutions, and this solution is unique. These are all the contents of this thesis.

【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O175

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