【摘要】：本文利用變分方法研究了一階Hamilton系統(tǒng)和幾類二階阻尼微分方程的同宿軌道和異宿軌道問(wèn)題.在各種假設(shè)條件下,分別獲得了同宿軌道和異宿軌道的存在性和多解性.主要內(nèi)容如下：第一章介紹一些研究的背景、研究概況以及一些預(yù)備知識(shí).第二章考慮如下的一階Hamilton系統(tǒng)z=JHz(t,z),a.e.t∈R,這里的H(t,z)依賴于t,但關(guān)于t不是周期的.在一些比著名的(Ambrosetti-Rabinowitz)超二次條件(簡(jiǎn)稱為(AR)條件)弱的超二次假設(shè)下,利用環(huán)繞定理得到了同宿軌道的存在性.另外,還討論了次二次條件下同宿軌道的多解性.改進(jìn)和推廣了一些文獻(xiàn)中的已有結(jié)果.第三章討論如下帶阻尼的微分方程ü+cu-L(t)u+Wu_(t,u)=0,其中c≥0是一個(gè)常數(shù)；對(duì)稱矩陣L(t)關(guān)于t是非周期的；W(t,u)依賴于t,但關(guān)于t不是周期的.在W(t,u)為次二次或者超二次情形下,利用臨界點(diǎn)理論得到了該方程的無(wú)窮多個(gè)同宿軌道或擬同宿軌道的存在性.改進(jìn)了現(xiàn)有文獻(xiàn)中的一些結(jié)果,同時(shí)對(duì)于Zhang,Yuan提出的公開(kāi)問(wèn)題給出了明確的回答.此外,利用Nehari流形還考慮了當(dāng)W(t,u)不定號(hào)時(shí)該方程的同宿軌道或擬同宿軌道的存在性.第四章討論如下帶阻尼的微分方程ü+g(t)u-L(t)u+Wu_(t,u)=0,其中g(shù)∈C(R, R);對(duì)稱矩陣L(t)關(guān)于t不是周期的；W(t,u)依賴于t,但關(guān)于t不是周期的.在關(guān)于g,L以及W的一些合理假設(shè)下,利用噴泉定理和對(duì)偶的噴泉定理討論了當(dāng)W(t,u)分別是超二次、次二次以及凹凸組合項(xiàng)情形時(shí),該方程無(wú)窮多個(gè)同宿軌道的存在性問(wèn)題,改進(jìn)和推廣了現(xiàn)有文獻(xiàn)中的一些結(jié)果.第五章考慮如下方程的同宿軌道和異宿軌道問(wèn)題ü+Au-L(t)u+Wu_(t,u)=f(t),其中A是一個(gè)反對(duì)稱常數(shù)矩陣；L(t)∈C(R,RN2)是一個(gè)對(duì)稱一致正定矩陣；函數(shù)f∈L2(R,RN)且w∈C1(R×RN,R)首先,考慮了強(qiáng)不定問(wèn)題的同宿軌道的存在性和多解性,其中W(t,u)關(guān)于t是周期的且滿足較弱的漸進(jìn)二次條件.然后,在關(guān)于L(t),f(t)和W(t,u)的某些假設(shè)條件下,利用變分方法得到了異宿軌道的存在性.即對(duì)于某個(gè)子集m(?)RN, (?)x∈m都存在一個(gè)異宿軌道w,使得w(-∞)=x且w(+∞)∈m\{x}.
[Abstract]:In this paper, we use the variational method to study the homoclinic orbits of the first order Hamilton system and several classes of two order damped differential equations. Under various hypothetical conditions, the existence and multiple solutions of the homoclinic orbits are obtained respectively. The main contents are as follows: the first chapter introduces some research background, research situation and some of them. In the second chapter, the second chapter considers the following first order Hamilton system z=JHz (T, z) and a.e.t R, where H (T, z) depends on T, but t is not periodic. Under the hypothetical super two assumption that is weaker than the famous (Ambrosetti-Rabinowitz) two times condition, the existence of the homoclinic orbit is obtained by the surround theorem. In addition, the existence of the homoclinic orbit is obtained. We discuss the multiple solvability of homoclinic orbits under the two times. Improve and extend the existing results in some literature. The third chapter discusses the following differential equation u +cu-L (T) u+Wu_ (T, U) =0 with damping, where C > 0 is a constant; the symmetric matrix L (T) is non periodic on T; W (T,) is not periodic. In the case of two or two times, the existence of infinitely many homoclinic orbitals and quasi homoclinic orbitals of the equation is obtained by using the critical point theory. Some results in the existing literature are improved, and a clear answer to the open problem proposed by Zhang and Yuan is given. Furthermore, the use of the Nehari manifold is also considered when the W (T, U) indefinite number is considered. The existence of the homoclinic orbits of the equation. The fourth chapter discusses the following differential equations with damped +g (T) u-L (T) u+Wu_ (T, U) =0, wherein the G C (R, R); the symmetric matrix is not periodic; The fountain theorem discusses the existence of infinitely many homoclinic orbits when W (T, U) is super two, two times and concave convex combination terms, and improves and generalizes some results in the existing literature. The fifth chapter considers the homoclinic orbits of the following equation, the problem of u +Au-L (T) u+Wu_ (t, U) =f (T) of the following equation, and A is an inverse. A symmetric constant matrix; L (T) C (R, RN2) is a symmetric and consistent positive definite matrix; the function f L2 (R, RN) and w C1 C1 (R x), first, consider the existence and multi solvability of the homoclinic orbits of strongly indefinite problems. The existence of the heteroclinic orbit is obtained by the variational method. That is, there is a heteroclinic orbit w for a subset of M (?) RN and (?) x m, which makes w (- infinity) =x and w (+ infinity) m{x}. m{x}.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175

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