Ordinary Differential Equations in Rn : Problems and Methods

During the fifties, one of the authors, G. Stampacchia, had prepared some lecture notes on ordinary differential equations for a course in ad­ analysis. These remained for a long time unused because he was no vanced longer very interested in the study of such equations. We now see, though, that numerous applications to biology, chemistry, economics, and medicine have recently been added to the traditional ones in mechanics; also, there has been in these last years a reemergence of interest in nonlinear analy­ sis, of which the theory of ordinary differential euqations is one of the principal sources of methods and problems. Hence the idea to write a book. 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. The book has been conceived in such a way so that even the reader who has merely had a first course in calculus may be able to study it and to obtain a panoramic vision of the theory. We have tried to avoid abstract formalism, preferring instead a discursive style, which should make the book accessible to engineers and physicists without specific preparation in modern mathematics. For students of mathematics, it pro­ vides motivation for the subject of more advanced analysis courses.

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.