常微分方程基礎理論（影印版）

前言

為了更好地借鑒國外數(shù)學教育與研究的成功經(jīng)驗，促進我國數(shù)學教育與研究事業(yè)的發(fā)展，提高高等學校數(shù)學教育教學質(zhì)量，本著“為我國熱愛數(shù)學的青年創(chuàng)造一個較好的學習數(shù)學的環(huán)境”這一宗旨，天元基金贊助出版“天元基金影印數(shù)學叢書”。該叢書主要包含國外反映近代數(shù)學發(fā)展的純數(shù)學與應用數(shù)學方面的優(yōu)秀書籍，天元基金邀請國內(nèi)各個方向的知名數(shù)學家參與選題的工作，經(jīng)專家遴選、推薦，由高等教育出版社影印出版。為了提高我國數(shù)學研究生教學的水平，暫把選書的目標確定在研究生教材上。當然，有的書也可作為高年級本科生教材或參考書，有的書則介于研究生教材與專著之間。歡迎各方專家、讀者對本叢書的選題、印刷、銷售等工作提出批評和建議。

內(nèi)容概要

　　本書為開展常微分方程研究工作的讀者提供必要的準備知識，可作為本科高年級和研究生常微分方程課程教材。　　本書內(nèi)容分為四部分：第一部分(第一、二、三章)的內(nèi)容包括解的存在性、唯一性、對數(shù)據(jù)的光滑依賴性，以及解的非唯一性；第二部分(第四、六、七章)討論線性常微分方程，書中用矩陣的S-N分解代替Jordan分解，前者的計算較后者更容易；第三部分(第八、九、十章)討論非線性常微分方程的穩(wěn)定性、漸近穩(wěn)定性等幾何理論；第四部分(第五、十一，十二、十三章)討論常微分方程的冪級數(shù)解，包括線性常微分方程的奇點分類及非線性常微分方程當參數(shù)或自變量趨向某奇點時的漸近解等。

書籍目錄

PrefaceChapter Ⅰ.Fundamental Theorems of Ordinary Differential Equations?、?1.Existence and uniqueness with the Lipschitz condition?、?2.Existence without the Lipschitz condition?、?3.Some global properties of solutions?、?4.Analytic differential equations　Exercises ⅠChapterⅡ.Dependence on Data?、?1.Continuity with respect to initial data and parameters?、?2.Differentiability　Exercises ⅡChapter Ⅲ.Nonuniqueness?、?1.Examples?、?2.The Kneser theorem　Ⅲ-3.Solution curves on the boundary of R(A)?、?4.Maximal and minimal solutions?、?5.A comparison theorem?、?6.Sufficient conditions for uniqueness　Exercises ⅢChapter Ⅳ.General Theory of Linear Systems?、?1.Some basic results concerning matrices?、?2.Homogeneous systems of linear differential equations?、?3.Homogeneous systems with constant coefficients?、?4.Systems with periodic coefficients　Ⅳ-5.Linear Hamiltonian systems with periodic coefficients?、?6.Nonhomogeneous equations?、?7.Higher-order scalar equations　Exercises ⅣChapter Ⅴ.Singularities of the First Kind　Ⅴ-1.Formal solutions of an algebraic differential equation?、?2.Convergence of formal solutions of a system of the first kind　Ⅴ-3.The S-N decomposition of a matrix of infinite order?、?4.The S-N decomposition of a differential operator?、?5.A normal form of a differential operator?、?6.Calculation of the normal form of a differential operator?、?7.Classification of singularities of homogeneous linear systems　Exercises ⅤChapter Ⅵ.Boundary-Value Problems of Linear Differential Equations of the Second-Order?、?1.Zeros of solutions?、?2.Sturm-Liouville problems?、?3.Eigenvalue problems?、?4.Eigenfunction expansions　Ⅵ-5.Jost solutions?、?6.Scattering data?、?7.Refiectionless potentials　Ⅵ-8.Construction of a potential for given data?、?9.Differential equations satisfied by reflectionless potentials?、?10.Periodic potentials　Exercises ⅥChapter Ⅶ.Asymptotic Behavior of Solutions of Linear Systems?、?1.Liapounoff's type numbers?、?2.Liapounoff's type numbers of a homogeneous linear system　Ⅶ-3.Calculation of Liapounoff's type numbers of solutions?、?4.A diagonalization theorem?、?5.Systems with asymptotically constant coefficients?、?6.An application of the Floquet theorem　Exercises ⅦChapter Ⅷ.Stability　Ⅷ-1.Basic definitions　Ⅷ-2.A sufficient condition for asymptotic stability?、?3.Stable manifolds?、?4.Analytic structure of stable manifolds?、?5.Two-dimensional linear systems with constant coefficients　Ⅷ-6.Analytic systems in R2?、?7.Perturbations of an improper node and a saddle point?、?8.Perturbations of a proper node?、?9.Perturbation of a spiral point?、?10.Perturbation of a center　Exercises ⅧChapter Ⅸ.Autonomous Systems　Ⅸ-1.Limit-invariant sets?、?2.Liapounoff's direct method?、?3.Orbital stability　Ⅸ-4.The Poincare-Bendixson theorem?、?5.Indices of Jordan curves　Exercises ⅨChapter Ⅹ.The Second-Order Differential Equation (d2x)/(dt2)+h(x)*(dx)/(dt)+g(x)=0?、?1.Two-point boundary-value problems?、?2.Applications of the Liapounoff functions　Ⅹ-3.Existence and uniqueness of periodic orbits?、?4.Multipliers of the periodic orbit of the van der Pol equation?、?5.The van der Pol equation for a small ε > 0　Ⅹ-6.The van der Pol equation for a large parameter?、?7.A theorem due to M.Nagumo　Ⅹ-8.A singular perturbation problem　Exercises ⅩChapter Ⅺ.Asymptotic Expansions?、?1.Asymptotic expansions in the sense of Poincare　Ⅺ-2.Gevrey asymptotics?、?3.Flat functions in the Gevrey asymptotics?、?4.Basic properties of Gevrey asymptotic expansions?、?5.Proof of Lemma Ⅺ-2-6　Exercises ⅪChapter Ⅻ.Asymptotic Expansions in a Parameter?、?1.An existence theorem　Ⅻ-2.Basic estimates?、?3.Proof of Theorem Ⅻ-1-2?、?4.A block-diagonalization theorem?、?5.Gevrey asymptotic solutions in a parameter　Ⅻ-6.Analytic simplification in a parameter　Exercises ⅫChapter ⅩⅢ.Singularities of the Second Kind?、?1.An existence theorem?、?2.Basic estimates　ⅩⅢ-3.Proof of Theorem ⅩⅢ-1-2?、?4.A block-diagonalization theorem?、?5.Cyclic vectors (A lemma of P.Deligne)?、?6.The Hukuhara-Turrittin theorem?、?7.An n-th-order linear differential equation at a singular point of the second kind?、?8.Gevrey property of asymptotic solutions at an irregular singular pointExercises ⅩⅢReferencesIndex

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《常微分方程基礎理論(影印版)》的引進是為了更好地借鑒國外微積分教學與研究的成功經(jīng)驗，促進我國數(shù)學教育與研究事業(yè)的發(fā)展，提高高等學校數(shù)學教育教學質(zhì)量，為本科高年級和研究生開展常微分程研究工作提供必要的理論依據(jù)，《常微分方程基礎理論(影印版)》為原版影印，既可供本科高年級和研究生自學參考，也可做為教材使用。

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