本文選題：周期解 + 平均方法　； 參考：《吉林大學(xué)》2017年博士論文

【摘要】：眾所周知,現(xiàn)實世界中的許多現(xiàn)象都具有周期性.自法國數(shù)學(xué)家Poincare和俄國數(shù)學(xué)家Lyapunov以來對于連續(xù)動力系統(tǒng)的周期解存在性的研究一直是動力系統(tǒng)研究的中心課題之一.然而,并非所有的自然現(xiàn)象都能用連續(xù)的系統(tǒng)或者離散的系統(tǒng)來描述.目前,一些連續(xù)系統(tǒng)的理論和方法己經(jīng)發(fā)展到了時標上,例如[10,78,55,53].時標是R中的任意非空閉子集,通常表示為T.時標理論的建立主要是為了研究連續(xù)和離散混合的系統(tǒng).例如,如果T = Z,時標上的動態(tài)方程表現(xiàn)為差分方程的形式,相反,如果T = R,時標上的動態(tài)方程則表現(xiàn)為微分方程的形式.所以說時標理論統(tǒng)一且推廣了現(xiàn)有的微分和差分理論.時標理論在經(jīng)濟學(xué),人口模型,生物模型等方面都有重要的應(yīng)用.隨著這一理論的迅速發(fā)展,對于時標上動態(tài)方程周期解的研究也吸引了越來越多人們的關(guān)注.到目前為止,當“時間”是連續(xù)和離散混合情況的帶有小參數(shù)ε擾動的動態(tài)方程周期解的存在性問題正逐漸被關(guān)注.本篇博士論文主要研究的是下面的時標上帶有擾動的動態(tài)方程周期解的存在性問題其中fi:T×U→Rn,i=0,1,…,k和r:T×U×(—ε0,ε0)→Rn是rd-連續(xù)的函數(shù),關(guān)于t是T-周期的,U是Rn中的開集,ε是一個小參數(shù).在本篇文章中我們主要應(yīng)用重合度理論和平均法去研究時標上帶有擾動的動態(tài)方程的周期解的存在性.同時我們將時標上帶有擾動的動態(tài)方程的平均法進一步的推廣到ε的任意階.具體地說,我們是應(yīng)用拓撲度理論去證明時標上動態(tài)方程周期解的存在性.我們的證明受連續(xù)系統(tǒng)經(jīng)典結(jié)果的啟發(fā),但是在時標上的證明過程會更復(fù)雜.本篇博士論文總共分三章,第一章為緒論,第二章和第三章為主要結(jié)果.在第一章中,我們討論了微分方程周期解的發(fā)展歷史,平均方法的起源和發(fā)展,以及時標理論的起源和發(fā)展,并且簡單介紹了我們所考慮問題的研究背景以及概括了本文的主要工作.然后,我們簡單介紹了時標的定義,時標理論的相關(guān)概念及在本文證明中將用到的時標上的結(jié)果.在最后一節(jié)中,我們總結(jié)了本文的主要工作.在第二章中,我們給出了本文的第一個主要結(jié)果,即用重合度理論證明時標上動態(tài)方程周期解的存在性定理.在本章的第一節(jié)中,我們給出了一階的周期解存在性定理.然后關(guān)于n階擾動的時標上的動態(tài)方程,我們進而給出了任意階的周期解存在性定理.最近,Llibre,Novaes和Teixeira將帶擾動的非線性微分方程的平均法推廣到了 ε的任意階,他們的結(jié)果主要是應(yīng)用Brouwer度理論證明的,與他們的結(jié)果相比,我們的結(jié)果把連續(xù)時間上的平均定理建立在時標上,同時我們給出了新的條件,并且給出了更容易計算的平均函數(shù),我們也不要求動態(tài)方程中的函數(shù)是充分光滑的.事實上,我們的主要結(jié)果提供了一種用拓撲方法探究時標上動態(tài)方程周期解存在性的理論.接下來,我們可以給出判定時標上ε任意階的非線性動態(tài)方程的周期解存在的主要結(jié)果.定理0.0.1設(shè)T是一個T-周期時標,U(?)Rn是一個有界開集.考慮下面的動態(tài)方程其中 fi:T× U→ Rn,i=1,…,k,r:T×U/×(-ε0,ε0)都是 rd-連續(xù)的函數(shù),關(guān)于t是T-周期的,并且關(guān)于x是局部Lipschitz的.并且,我們作如下假設(shè):(i)對于任意的t ∈ T,p ∈ 存在p的一個鄰域Np,與ε獨立的一個常數(shù)σ0和整數(shù)1≤j ≤ n,對任意的q ∈ Np,t ∈[0,T]T和ε ∈[—ε0,ε0]\{0}}滿足(ii)假設(shè)對任意的其中平均函數(shù)如果對于方程(0.0.2),(i)和(ii)始終成立,則方程(0.0.2)存在一個T—周期解x(t),滿足對于任意的t ∈ T,充分小的|ε|0,都有x(t)∈U.利用拓撲度理論,在2.2節(jié)中我們將會給出該定理的證明.最后,為了說明我們的主要結(jié)果,在該章的最后一節(jié)我們給出了一些例子.在第三章中,我們將Llibre的連續(xù)時間上的平均定理推廣到時標上.在Llibre之前的研究中,他已經(jīng)用低階平均定理證明了擾動的微分方程的周期解的存在性.本章的主要結(jié)果把Llibre的微分方程的平均定理建立到時標上的動態(tài)方程上.首先,在3.1節(jié)和3.2節(jié)中我們給出了時標上動態(tài)方程的一階平均定理和二階平均定理.進而在3.3節(jié)中我們給出了證明時標上動態(tài)方程的周期解存在的任意階的平均定理.我們定義時標上的i階平均函數(shù)Fi如下:其中定義yi:T × → Rn,i = 1,2,…,k-1滿足下面的迭代積分方程,其中= b1 + b2+…+ bl,Sl表示所有非負整數(shù)的l-元組(b1,b2,…,bl)的集合,并且(b1,b2,…,bl)是Diophantine方程b1 + 2b2 + … + lbl = l的所有非負整數(shù)解.接下來我們分兩種情況陳述高階平均定理:當f0 = 0時是定理0.0.2;當f0 ≠ 0時是定理0.0.3.定理0.0.2假設(shè)f0 = 0,考慮時標上的動態(tài)方程其中 fi:T × U→Rn,i = 1,…,k 和 r:T×U(-ε0,ε0)→ Rn 都是 rd-連續(xù)的函數(shù),關(guān)于t是T-周期的,U是Rn中的一個開集,ε是一個小參數(shù).我們假設(shè)其滿足下面的條件:(ⅰ)對于任意的t ∈ T,fi(t,·)∈ Ck,i = 1,2,…,k.函數(shù)(?)kfi和r關(guān)于第二個變量是局部Lipschitz的.(ⅱ)存在r ∈{1,2,…,k},有Fr≠0,假設(shè)Fi=0,i=0,1,2,…,r-1(這里我們?nèi)0=0).(ⅲ)假設(shè)存在某個a ∈ U有Fr(a)= 0,對于a的鄰域V(?)U滿足當z ∈ V-{a}時有Fr(z)≠ 0,并且Brouwer度度deg(Fr(z),V,0)≠0.那么,對于充分小的|ε|0,方程(0.0.5)存在一個T-周期解x(·,ε)滿足當 ε → 0 時,x(·,ε)→a.定理0.0.3假設(shè)f0 ≠ 0,考慮時標上的動態(tài)方程其中 fi:T × U→ Rn,i = 0,1,…,k和 r:T × U ×(-ε0,ε0)→ Rn 都是連續(xù)的函數(shù),關(guān)于t是T-周期的,U是Rn中的一個開集,ε是一個小參數(shù).我們假設(shè)其滿足下面的條件:(i'),假設(shè)存在U的一個開子集W滿足對任意的z ∈W,都有φ(t,z)是T-周期的.(ii')對于任意的t ∈ T,fi(t,·)∈ Ck,i = 0,1,2,…,函數(shù)(?)kfi,i =0,1,2,…,k和r關(guān)于第二個變量是局部Lipschitz的.(iii')存在r∈{1,2,…,k},有Fr≠0,假設(shè)Fi=0,i = 0,1,2,…,r-1.與此同時,我們假設(shè)存在某個a ∈ W有Fr(a)= 0,對于a的鄰域V(?)W滿足當z ∈ V-{a}時有Fr(z)≠ 0,并且Brouwer度deg(Fr(z),V,0)≠ 0.那么,對于充分小的|ε|0,方程(0.0.6)存在一個T-周期解x(·,ε)滿足當 ε → 0 時,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.
【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】：2017
【分類號】：O175

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