Sinopsis
I Existence and Uniqueness for the Initial Value Problem Under the Hypothesis of Lipschitz.- 1. General Results.- 1.1 Definitions.- 1.2 Geometrical Interpretation.- 1.3 Functions Satisfying a Lipschitz Condition.- 1.4 Existence Theorem.- 1.5 Uniqueness Theorem.- 1.6 Continuous Dependence on Initial Conditions and Parameters.- 1.7 Interval of Definition and Extension of Solutions.- 1.8 Gronwall's Lemma.- 1.9 Application of Gronwall's Lemma to the Cauchy Problem.- 2. Qualitative Properties of Solutions.- 2.1 Differentiability of Solutions.- 2.2 Analyticity of the Solutions.- 3. Solutions as Functions of the Initial Data.- 3.1 Differentiability with Respect to the Parameter.- 3.2 Differentiability with Respect to the Initial Point.- 3.3 Higher Order Differentiability and Analyticity.- 3.4 Remark about a More General Point of View.- 4. Systems of Equations as Particular Transformations Between Function Spaces.- 4.1 Review of Metric Spaces.- 4.2 Review of Banach Spaces.- 4.3 The Cauchy Problem and Fixed Points of Certain Transformations in Banach Spaces.- 5. Exercises.- 5.1 Variables Separable Equations.- 5.2 Equations Reducible to Separable Equations.- 5.3 Linear Equations of the First Order.- 5.4 Linear Equations of Order Higher than the First with Constant Coefficients.- 5.5 Euler Equations.- 5.6 Envelopes and Differential Equations.- 5.7 Various Exercises.- 5.8 Selected Exercises.- 6. Bibliographical Notes.- II Linear Systems.- 1. Elements of Linear Algebra.- 1.1 Matrices and Eigenvalues.- 1.2 Linear Operators Between Banach Spaces.- 1.3 Canonical Form of Matrices.- 1.4 Spectrum and Eigenvalues of a Linear Operator.- 1.5 Limits of Operators.- 2. Linear Systems of Ordinary Differential Equations.- 2.1 Formal Solution of Linear Systems.- 2.2 Fundamental Systems of Solutions and Adjoint Systems.- 2.3 Nonhomogeneous Systems.- 3. Operational Calculus.- 3.1 Analytic Functions of Operators.- 3.2 Linear Systems with Constant Coefficients.- 4. Linear Finite Differences Equations.- 4.1 Homogeneous Linear Finite Differences Equations.- 4.2 Nonhomogeneous Linear Finite Differences Equations.- 5. Examples.- 6. Bibliography.- III Existence and Uniqueness for the Cauchy Problem Under the Condition of Continuity.- 1. Existence Theorem.- 1.1 Characterization of Compact Sets of Continuous Functions: Ascoli's Theorem.- 1.2 Local Existence.- 1.3 Global Existence.- 2. The Peano Phenomenon.- 2.1 Approximation of all Solutions to a Given Cauchy Problem.- 2.2 Maximal and Minimal Solutions. The Peano Phenomenon.- 2.3 The Peano Phenomenon for Systems.- 2.4 Maximal Solutions, Differential Inequalities, and Global Existence.- 3. Questions of Uniqueness.- 3.1 Continuous Dependence.- 3.2 Uniqueness Theorems.- 3.3 How Many Differential Equations Have the Uniqueness Property?.- 4. Elements of G-Convergence.- 4.1 Introduction.- 4.2 G-Convergence for Equations Satisfying the Lipschitz Condition.- 4.3 Homogenization.- 4.4 G-Compactness.- 4.5 G-Convergence and the Peano Phenomenon.- 5. Bibliographical Notes.- IV Boundary Value Problems.- 1. Continuous Mappings on Euclidean Spaces.- 1.1 The Topological Degree.- 1.2 The Theorems of Brouwer and Miranda.- 2. Geometric Boundary Value Problems.- 2.1 The Boundary Value Problems of Picard and Nicoletti.- 2.2 A Geometrical Formulation of the Boundary Value Problem.- 2.3 Some Applications of the Geometric Formulation.- 3. Sturm-Liouvilie Problems: Eigenvalues and Existence and Uniqueness Theorems.- 3.1 Eigenvalues and Eigenfunctions.- 3.2 Prüfer's Change of Variables.- 3.3 Existence and Properties of the Eigenvalues.- 3.4 Applications to Questions of Uniqueness for Problems Involving Nonlinear Equations.- 3.5 Application to the Existence of Solutions for Problems Involving Nonlinear Equations.- 3.6 Further Properties of Eigenvalues and Eigenfunctions.- 4. Periodic Solutions.- 4.1 The Case of First Order Equations.- 4.2 The Case of Second Order Equations.- 4.3 The Case of Systems.- 4.4 On the Structure of Periodic Solu
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