During the fifties, one of the authors, G. Stampacchia, had prepared some lecture notes on ordinary differential equations for a course in adยญ 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 analyยญ 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 proยญ vides motivation for the subject of more advanced analysis courses
CONTENT
I Existence and Uniqueness for the Initial Value Problem Under the Hypothesis of Lipschitz -- 1. General Results -- 2. Qualitative Properties of Solutions -- 3. Solutions as Functions of the Initial Data -- 4. Systems of Equations as Particular Transformations Between Function Spaces -- 5. Exercises -- 6. Bibliographical Notes -- II Linear Systems -- 1. Elements of Linear Algebra -- 2. Linear Systems of Ordinary Differential Equations -- 3. Operational Calculus -- 4. Linear Finite Differences Equations -- 5. Examples -- 6. Bibliography -- III Existence and Uniqueness for the Cauchy Problem Under the Condition of Continuity -- 1. Existence Theorem -- 2. The Peano Phenomenon -- 3. Questions of Uniqueness -- 4. Elements of G-Convergence -- 5. Bibliographical Notes -- IV Boundary Value Problems -- 1. Continuous Mappings on Euclidean Spaces -- 2. Geometric Boundary Value Problems -- 3. Sturm-Liouvilie Problems: Eigenvalues and Existence and Uniqueness Theorems -- 4. Periodic Solutions -- 5. Functional Boundary Value Problems -- 6. Bibliographical Notes -- V Questions of Stability -- 1. Stability of the Solutions of Linear Systems -- 2. Some Methods for the Determination of the Stability of Nonlinear Systems -- 3. Some Applications -- 4. The Method of Runge and Kutta -- 5. Bibliographical Notes
