时标上动态方程周期解的存在性

发布时间：2018-07-01 16:49

本文选题：周期解 + 平均方法　；参考：《吉林大学》2017年博士论文

【摘要】：众所周知,现实世界中的许多现象都具有周期性.自法国数学家Poincare和俄国数学家Lyapunov以来对于连续动力系统的周期解存在性的研究一直是动力系统研究的中心课题之一.然而,并非所有的自然现象都能用连续的系统或者离散的系统来描述.目前,一些连续系统的理论和方法己经发展到了时标上,例如[10,78,55,53].时标是R中的任意非空闭子集,通常表示为T.时标理论的建立主要是为了研究连续和离散混合的系统.例如,如果T = Z,时标上的动态方程表现为差分方程的形式,相反,如果T = R,时标上的动态方程则表现为微分方程的形式.所以说时标理论统一且推广了现有的微分和差分理论.时标理论在经济学,人口模型,生物模型等方面都有重要的应用.随着这一理论的迅速发展,对于时标上动态方程周期解的研究也吸引了越来越多人们的关注.到目前为止,当“时间”是连续和离散混合情况的带有小参数ε扰动的动态方程周期解的存在性问题正逐渐被关注.本篇博士论文主要研究的是下面的时标上带有扰动的动态方程周期解的存在性问题其中fi:T×U→Rn,i=0,1,…,k和r:T×U×(—ε0,ε0)→Rn是rd-连续的函数,关于t是T-周期的,U是Rn中的开集,ε是一个小参数.在本篇文章中我们主要应用重合度理论和平均法去研究时标上带有扰动的动态方程的周期解的存在性.同时我们将时标上带有扰动的动态方程的平均法进一步的推广到ε的任意阶.具体地说,我们是应用拓扑度理论去证明时标上动态方程周期解的存在性.我们的证明受连续系统经典结果的启发,但是在时标上的证明过程会更复杂.本篇博士论文总共分三章,第一章为绪论,第二章和第三章为主要结果.在第一章中,我们讨论了微分方程周期解的发展历史,平均方法的起源和发展,以及时标理论的起源和发展,并且简单介绍了我们所考虑问题的研究背景以及概括了本文的主要工作.然后,我们简单介绍了时标的定义,时标理论的相关概念及在本文证明中将用到的时标上的结果.在最后一节中,我们总结了本文的主要工作.在第二章中,我们给出了本文的第一个主要结果,即用重合度理论证明时标上动态方程周期解的存在性定理.在本章的第一节中,我们给出了一阶的周期解存在性定理.然后关于n阶扰动的时标上的动态方程,我们进而给出了任意阶的周期解存在性定理.最近,Llibre,Novaes和Teixeira将带扰动的非线性微分方程的平均法推广到了 ε的任意阶,他们的结果主要是应用Brouwer度理论证明的,与他们的结果相比,我们的结果把连续时间上的平均定理建立在时标上,同时我们给出了新的条件,并且给出了更容易计算的平均函数,我们也不要求动态方程中的函数是充分光滑的.事实上,我们的主要结果提供了一种用拓扑方法探究时标上动态方程周期解存在性的理论.接下来,我们可以给出判定时标上ε任意阶的非线性动态方程的周期解存在的主要结果.定理0.0.1设T是一个T-周期时标,U(?)Rn是一个有界开集.考虑下面的动态方程其中 fi:T× U→ Rn,i=1,…,k,r:T×U/×(-ε0,ε0)都是 rd-连续的函数,关于t是T-周期的,并且关于x是局部Lipschitz的.并且,我们作如下假设:(i)对于任意的t ∈ T,p ∈ 存在p的一个邻域Np,与ε独立的一个常数σ0和整数1≤j ≤ n,对任意的q ∈ Np,t ∈[0,T]T和ε ∈[—ε0,ε0]\{0}}满足(ii)假设对任意的其中平均函数如果对于方程(0.0.2),(i)和(ii)始终成立,则方程(0.0.2)存在一个T—周期解x(t),满足对于任意的t ∈ T,充分小的|ε|0,都有x(t)∈U.利用拓扑度理论,在2.2节中我们将会给出该定理的证明.最后,为了说明我们的主要结果,在该章的最后一节我们给出了一些例子.在第三章中,我们将Llibre的连续时间上的平均定理推广到时标上.在Llibre之前的研究中,他已经用低阶平均定理证明了扰动的微分方程的周期解的存在性.本章的主要结果把Llibre的微分方程的平均定理建立到时标上的动态方程上.首先,在3.1节和3.2节中我们给出了时标上动态方程的一阶平均定理和二阶平均定理.进而在3.3节中我们给出了证明时标上动态方程的周期解存在的任意阶的平均定理.我们定义时标上的i阶平均函数Fi如下:其中定义yi:T × → Rn,i = 1,2,…,k-1满足下面的迭代积分方程,其中= b1 + b2+…+ bl,Sl表示所有非负整数的l-元组(b1,b2,…,bl)的集合,并且(b1,b2,…,bl)是Diophantine方程b1 + 2b2 + … + lbl = l的所有非负整数解.接下来我们分两种情况陈述高阶平均定理:当f0 = 0时是定理0.0.2;当f0 ≠ 0时是定理0.0.3.定理0.0.2假设f0 = 0,考虑时标上的动态方程其中 fi:T × U→Rn,i = 1,…,k 和 r:T×U(-ε0,ε0)→ Rn 都是 rd-连续的函数,关于t是T-周期的,U是Rn中的一个开集,ε是一个小参数.我们假设其满足下面的条件:(ⅰ)对于任意的t ∈ T,fi(t,·)∈ Ck,i = 1,2,…,k.函数(?)kfi和r关于第二个变量是局部Lipschitz的.(ⅱ)存在r ∈{1,2,…,k},有Fr≠0,假设Fi=0,i=0,1,2,…,r-1(这里我们取F0=0).(ⅲ)假设存在某个a ∈ U有Fr(a)= 0,对于a的邻域V(?)U满足当z ∈ V-{a}时有Fr(z)≠ 0,并且Brouwer度度deg(Fr(z),V,0)≠0.那么,对于充分小的|ε|0,方程(0.0.5)存在一个T-周期解x(·,ε)满足当 ε → 0 时,x(·,ε)→a.定理0.0.3假设f0 ≠ 0,考虑时标上的动态方程其中 fi:T × U→ Rn,i = 0,1,…,k和 r:T × U ×(-ε0,ε0)→ Rn 都是连续的函数,关于t是T-周期的,U是Rn中的一个开集,ε是一个小参数.我们假设其满足下面的条件:(i'),假设存在U的一个开子集W满足对任意的z ∈W,都有φ(t,z)是T-周期的.(ii')对于任意的t ∈ T,fi(t,·)∈ Ck,i = 0,1,2,…,函数(?)kfi,i =0,1,2,…,k和r关于第二个变量是局部Lipschitz的.(iii')存在r∈{1,2,…,k},有Fr≠0,假设Fi=0,i = 0,1,2,…,r-1.与此同时,我们假设存在某个a ∈ W有Fr(a)= 0,对于a的邻域V(?)W满足当z ∈ V-{a}时有Fr(z)≠ 0,并且Brouwer度deg(Fr(z),V,0)≠ 0.那么,对于充分小的|ε|0,方程(0.0.6)存在一个T-周期解x(·,ε)满足当 ε → 0 时,x(·,ε)→ a。
[Abstract]:As we all know, many phenomena in the real world are periodic. The study of the existence of periodic solutions for continuous dynamical systems since Poincare and Russian mathematician Lyapunov has been one of the central topics in the research of dynamic systems. However, not all natural phenomena can be used in continuous systems or discrete systems. At present, some theories and methods of continuous systems have been developed to the time scale, for example, [10,78,55,53]. time scales are any non empty closed subsets in R, and usually the establishment of T. time scale theory is mainly to study continuous and discrete mixed systems. For example, if T = Z, the dynamic equation on the time scale is a differential equation. On the contrary, if T = R, the dynamic equation on the time scale is expressed as a form of differential equation. So the theory of time scale is unified and extends the existing differential and difference theory. The time scale theory has important applications in economics, population model, biological model and so on. With the rapid development of this theory, the dynamic Fang Chengzhou is on the time scale. The study of periodic solutions has attracted more and more people's attention. So far, the existence of periodic solutions of dynamic equations with small parameter epsilon perturbations when "time" is a continuous and discrete mixing situation is becoming more and more concerned. In sexual problems, fi:T * U - Rn, i=0,1,... K and r:T x U x (- E 0, epsilon 0) - Rn are rd- continuous functions, t is a T- period, U is an open set in Rn, and the epsilon is a small parameter. In this article, we mainly use the coincidence degree theory and the mean method to study the existence of the periodic solution of the dynamic equation with the disturbance. At the same time, we will mark the dynamic equation with disturbance. The mean method is further extended to the arbitrary order of the epsilon. Specifically, we apply the topological degree theory to prove the existence of the periodic solution of the dynamic equation on the time scale. Our proof is inspired by the classical results of the continuous system, but the proof process on the time scale will be more complex. This paper is divided into three chapters, the first chapter is the introduction, and the first chapter is the introduction. The two and third chapters are the main results. In the first chapter, we discuss the history of the development of periodic solutions for differential equations, the origin and development of the mean method, the origin and development of the time scale theory, and briefly introduce the research background of the problems we consider and summarize the main work of this article. Then, we briefly introduce the time scale. In the last section, we summarize the main work of this paper. In the second chapter, we give the first main result of this paper, that is, the existence theorem of the periodic solution of the dynamic equation with the coincidence degree theory. In the one section, we give the first order existence theorem of periodic solution. Then, on the dynamic equation of the time scale of the n order perturbation, we then give the existence theorem of the periodic solution of arbitrary order. Recently, Llibre, Novaes and Teixeira extend the mean method of the nonlinear differential equation with perturbation to any order of the epsilon. Their results are mainly By using the Brouwer degree theory, compared with their results, our results set the mean theorem on the continuous time on the time scale, and we give new conditions, and give the more easy to calculate the average function, and we do not require that the number of functions in the dynamic equation is fully smooth. In fact, our main results are proposed. A theory for exploring the existence of periodic solutions of dynamic equations on time scales by topological methods is provided. Next, we can give the main results of the existence of periodic solutions for a nonlinear dynamic equation of an arbitrary order on the time scale. Theorem 0.0.1 T is a T- periodic time scale, and U (?) Rn is a bounded open set. Consider the following dynamic equation in which fi:T X U - Rn, i=1,... K, r:T x U/ x (- epsilon 0, epsilon 0) are rd- continuous functions, t is a T- cycle, and X is a local Lipschitz. II hypothesis that if the average function of any of them is always established for the equation (0.0.2), (I) and (II), then the equation (0.0.2) has a T periodic solution x (T), which satisfies the arbitrary t T, and is fully small for the |0, we have the theorem of topological degree in the 2.2 section. Finally, to illustrate us In the last section of this chapter, we give some examples. In the third chapter, we generalize the mean theorem on the continuous time of Llibre. In the study before Llibre, he has proved the existence of the periodic solution of the perturbation differential equation with the lower order mean theorem. The main result of this chapter is that the Llibre is the main result of this chapter. The mean theorem of differential equations establishes the dynamic equations on the time scale. First, in the 3.1 and 3.2 sections, we give the first order mean theorem and the two order mean theorem of the dynamic equation on the time scale. In the 3.3 section, we give the mean theorem of arbitrary order for the existence of the periodic solution of the dynamic equation on the time scale. The I order average function Fi is as follows: yi:T, * Rn, I = 1,2,... K-1 satisfies the following iterative integral equation, which = B1 + b2+... + BL, Sl represents all l- tuples of non negative integers (B1, B2,... The set of, BL) and (B1, B2,... BL) is the Diophantine equation B1 + 2B2 +... + LBL = l all non negative integer solutions. Next, we describe the high order mean theorem in two cases: when F0 = 0 is theorem 0.0.2; when F0 is 0, the theorem 0.0.3. theorem 0.0.2 assumes F0 = 0, considering the dynamic equation on the time scale, of which fi:T * U > Rn, I = 1,... K and r:T * U (- epsilon 0, epsilon 0) - Rn are all rd- continuous functions, t is a T- cycle, U is an open set in Rn, and the epsilon is a small parameter. The K. function (?) KFI and R about the second variable are local Lipschitz. (2) there is R {1,2 {1,2,... K}, Fr 0, suppose Fi=0, i=0,1,2,... R-1 (here we take F0=0). (III) suppose there is a certain a U that has Fr (a) = 0, and the neighborhood V (?) U satisfies the Z V-{a}. 0, consider the dynamic equations on time scales, where fi:T * U to Rn, I = 0,1,... K and r:T * U * (- epsilon 0, epsilon 0) - Rn are continuous functions, and the T is a T- cycle, U is an open set in Rn, and the epsilon is a small parameter. We assume that it satisfies the following condition: (I'), suppose there is an open subset of U that satisfies any Z. 1,2,... The function (?) KFI, I =0,1,2,... For K and R, the second variable is local Lipschitz. (III') there exists R {1,2 {1,2,... K}, Fr 0, suppose Fi=0, I = 0,1,2,... R-1., at the same time, we assume that there is a certain a W with Fr (a) = 0, and for the a neighborhood V (?) W to be a Z V-{a} there is Fr (z) 0.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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