Ordinary Differential Equations in Rn: Problems and Methods: 39 (Applied Mathematical Sciences, 39)

Vidossich, Giovanni,Stampacchia, Guido,Piccinini, Livio C.

ISBN 10: 0387907238 / ISBN 13: 9780387907239

Editore: Springer, 1984

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Dati bibliografici

• Titolo: Ordinary Differential Equations in Rn: Problems and Methods: 39 (Applied Mathematical Sciences, 39)
• Editore: Springer
• Data di pubblicazione: 1984
• Lingua: Inglese
• ISBN 10: 0387907238
• ISBN 13: 9780387907239
• Rilegatura: Paperback
• Condizione: Good

Descrizione articolo

Riassunto

Our text, based on the old notes and experience gained in many courses, seminars, and conferences, both in Italy and abroad, aims to give a simple and rapid introduction to the various themes, problems, and methods of the theory of ordinary differential equations.

Contenuti

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 Solutions.- 5. Functional Boundary Value Problems.- 5.1 Linear Functional Problems.- 5.2 Nonlinear Functional Problems.- 6. Bibliographical Notes.- V Questions of Stability.- 1. Stability of the Solutions of Linear Systems.- 1.1 Definition of Stability.- 1.2 Stability for Autonomous Linear Systems.- 1.3 Autonomous Linear Systems of the Second Order.- 1.4 Certain Stability Problems for Nonautonomous Linear Systems.- 2. Some Methods for the Determination of the Stability of Nonlinear Systems.- 2.1 Definitions.- 2.2 Liapunov’s Method.- 2.3 The Fixed Point Method: Asymptotic Equivalence.- 2.4 Olech’s Method.- 2.5 The Method of the Logarithmic Norm.- 2.6 Invariant Sets.- 3. Some Applications.- 3.1 Problems in Biology and Chemistry.- 3.2 Problems in Automatic Control Theory.- 4. The Method of Runge and Kutta.- 4.1 The Fourth Order Runge-Kutta Algorithm.- 4.2 Practical Use of the Runge-Kutta Method.- 5. Bibliographical Notes.