具指数二分性和指数三分性的微分方程的仿射周期解

发布时间：2018-04-25 16:48

本文选题：指数二分性 + 指数三分性　；参考：《吉林大学》2016年博士论文

【摘要】：微分方程的指数二分的概念最早是由Lyapunov和Poincare在19世纪末提出的,并在随后的时间里迅速成为了微分方程领域中的重要研究对象之一.1930年,Perron ([24])发展了线性微分方程的指数二分理论,并以其为工具研究了线性系统的条件稳定性问题.从那时起,指数二分的理论便在微分方程领域被数学家们广泛应用,其中的部分成果可以参见[12,13,25]及其相关的文献.在1974年,Sacker和Sell ([28])共同提出了指数三分的概念并建立了相关的基本理论.Elaydi和Hajek ([17])随后研究了微分系统的指数三分性.在之后的时间里,指数三分概念作为指数二分性质的推广,在动力系统相关领域的研究中同样发挥了重要的作用.指数二分性和指数三分性是定性理论中非常重要的渐近性.所以研究指数二分性和指数三分性是十分必要的.本文给出了微分方程的指数二分性和指数三分性及仿射周期解的深入研究.为了使得本文更加独立,我们在第一章中首先介绍了一些已有的工作,然后给出了要用到的仿射周期解的定义和一些性质,接着回顾了微分方程的指数二分性和指数三分性的概念,最后指出在本文中.我们以仿射周期系统为研究对象,讨论指数二分以及指数三分条件下仿射周期系统的仿射周期解以及伪仿射周期解的存在性问题.在第二章中,我们讨论了具有指数二分性的一阶仿射周期系统.首先,考虑了一阶线性非齐次方程x'=A(t)x+f(t),其中连续有界函数A(t):R1→Rn×Rn和f(t):R1→Rn满足(Q,T)-仿射周期性,若相应的齐次方程x'=A(t)x满足指数二分性,我们有如下的定理：定理1如果线性方程x'=A(t)x对于投影P是指数二分的,A(t)和f(t)满足(Q,T)-仿射周期性.那么非齐次线性微分方程x'=A(t)x+f(t)存在一个(Q,T)-仿射周期解.其次,考虑了一阶半线性微分方程x'=A(t)x+g(t,x(t)),其中g:R1×Rn→Rn连续,A(t)和g(t,x)是(Q,T)-仿射周期的,若相应的齐次方程x'=A(t)x满足指数二分性,我们有定理2线性微分方程x'=A(t)x对于投影映射P和正常数K,L,α,β是指数二分的.同时,假设A(t),g(t,x)是(Q,T)-仿射周期的,Q∈GL(n)如果g(t,x)是有界函数满足利普希茨条件,那么方程x'=A(t)x+g(t,x(t))存在唯一的(Q,T)-仿射周期解.事实上,定理2中关于g(t,x(t))的利普希茨条件可以由线性增长条件取代,具体定理如下：定理3如果线性微分方程x'=A(t)x对于投影映射P和常数K,L,α,β0是指数二分的.同时,假设A(t),g(t,x)是(Q,T)-仿射周期的,其中Q∈O(n),g(t,x)满足条件(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,我们讨论了具有指数二分性的高阶仿射周期系统.首先,考虑了n-维二阶线性非齐次仿射周期系统x"+p(t)x'+q(t)x= e(t),其中p(t),q(t):R1→Rn×n,e(t):R1→Rn是连续的、(Q,T)-仿射周期的,给出如下结果：定理4假设p(t),(qt)和e(t)是连续的(Q,T)-仿射周期函数,并且对于所有的t∈R1,F(t),G(t)是有界的.如果p(T)和q(t)满足下列条件之一：1).对于所有的t∈R1,p(t)和q(t)是正定或负定的；2).对于所有的t∈R1,q(t)是负定的,那么,对于所有的t∈R1,F(t)有k个实部小于等于-α(α0)和2n-k个实部大于等于β(β0)的特征值.进一步,假设(?)0εmin(α,β)存在δ=δ(α+β,ε)0使得：如果存在h0满足对于所有的|t2-t1|≤h,总有|F(t2)-F(t1)|≤δ,则方程x"+p(t)x'+q(t)x=e(t)存在(Q,T)-仿射周期解.其次,考虑了m阶线性非齐次系统x(m)=a(t)x+e(t),其中a(t):R1→Rn×n,e(t):R1→Rn连续,满足(Q,T)-仿射周期条件.具体地,我们给出了以下定理：定理5假设a(t)和e(t)是连续的(Q,T)-仿射周期函数,并且对于所有的t∈R1:A(t),G(t)是有界的.如果1).当m=4k,k∈Z时,对于所有的t∈R1:a(t)是负定的;2).当m=4k+2,k∈Z,对于所有的t∈R1,a(t)是正定的;3).当m=4k+1或4k+3,k∈Z,对于所有的t∈R1,a(t)是正定或负定的,那么,对于所有的t∈R1,A(t)有k个实部小于等于-α(α0)和mn-k个实部大于等于β(β0)的特征值.进一步,假设(?)0εmin(α,β),存在δ=δ(α+β,ε)0使得：如果存在h0满足对于所有的|t2-t1|≤h,总有|A(t2)-A(t1)|≤δ,则方程x(m)=a(t)x+e(t)存在(Q,T)-仿射周期解.对于指数三分性,我们在第四章中讨论了具有指数三分性的仿射周期系统.首先,考虑了半线性微分方程x'=A(t)x+g(t,x(t)),其中g:R1×Rn→Rn为连续函数,A(t)和g(t,x)为(Q,T)-仿射周期函数,其对应的齐次线性微分方程为x'=A(t)x.定理6如果方程x'=A(t)x对于投影P1,P2以及常数K,a是指数三分的.同时,A(t),g(t,x)是(Q,T)-仿射周期函数,g(t,x)是有界函数并且对任意的t,x,y,满足|g(t,x)-g(t,y)|≤N|x-y|,其中Q∈GL(n),N0是一个常数且使得下式成立则方程x'=A(t)x+g(t,x(t))存在唯一的(Q,T)-仿射周期解.接着,我们给出了伪仿射周期解的定义.对于伪(Q,T)-仿射周期解,我们同样可以证明下面的存在性定理.定理7对于系统x'=A(t)x+g(t,x(t)),如果A(t)是(Q,T)-仿射周期的,g(t,x)是一个可以分解为g(t,x)=g1(t,x)+g2(t,x)的伪仿射周期函数,其中Q∈GL(n),T0为常数,g1(t,x)∈CT, g2(t,x)∈C0同时,对于Rn的任意有界子集,函数g(t,x)和g1(t,x)均关于t∈R1一致连续.函数g(t,x)满足|g(t,x)-g(t,y)|≤N|x-y|,(?)t,x,y,其中N0为常数.如果系统x'=A(t)x+g(t,x(t))对应的齐次线性方程x'=A(t)x是指数三分的,并且指数三分条件中的投影Pl,P2以及常数K,α满足适当的条件,那么系统x'=A(t)x+g(t,x(t))一定存在伪(Q,T)-仿射周期解,并且这个解是唯一的.对于指数三分性,我们还有如下的推论：推论1如果方程x'=A(t)x+g(t,x(t))对于投影P1,P2以及常数K,α是指数三分的,同时,A(t),g(t,x)是(Q,T)-仿射周期函数,g(t,x)对于任意的t∈R1均关于x一致连续,并且满足|g(t,x)|≤a|x|+b,(?)t,x,其中Q∈O(n),a,b为大于0的常数,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.推论2对于系统x'=A(t)x+g(t,x(t)),假设A(t)是(Q,T)-仿射周期的,g(t,x)是一个可以分解为g(t,x)=g1(t,x)+g2(t,x)的伪仿射周期函数,其中Q∈O(n)(n),T0为常数g1(t,x)∈CT, g2(t,x)∈C0同时,g(t,x)对于任意的t∈R1均关于x一致连续,并且满足|g(t,x)|≤a|x|+b,(?)t,x,其中a,b为大于0的常数.如果系统x'=A(t)x+g(t,x(t))对应的齐次线性方程(1.4.13)是指数三分的,并且指数三分条件中的投影P1,P2以及常数K,α满足条件2Kα/α1,那么系统x'=A(t)x+g(t,x(t))一定存在伪(Q,T)-仿射周期解,并且这个解是唯一的.这些就是本论文的全部内容.
[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.

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

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