An Introduction to Ordinary Differential Equations

Ravi P. Agarwal, Donal O’Regan (auth.)978-0-387-71275-8, 978-0-387-71276-5

This textbook provides a rigorous and lucid introduction to the theory of ordinary differential equations (ODEs), which serve as mathematical models for many exciting real-world problems in science, engineering, and other disciplines.

Key Features of this textbook:

Effectively organizes the subject into easily manageable sections in the form of 42 class-tested lectures

Provides a theoretical treatment by organizing the material around theorems and proofs

Uses detailed examples to drive the presentation

Includes numerous exercise sets that encourage pursuing extensions of the material, each with an “answers or hints” section

Covers an array of advanced topics which allow for flexibility in developing the subject beyond the basics

Provides excellent grounding and inspiration for future research contributions to the field of ODEs and related areas

This book is ideal for a senior undergraduate or a graduate-level course on ordinary differential equations. Prerequisites include a course in calculus.

Series: Universitext

Ravi P. Agarwal received his Ph.D. in mathematics from the Indian Institute of Technology, Madras, India. He is a professor of mathematics at the Florida Institute of Technology. His research interests include numerical analysis, inequalities, fixed point theorems, and differential and difference equations. He is the author/co-author of over 800 journal articles and more than 20 books, and actively contributes to over 40 journals and book series in various capacities.

Donal O’Regan received his Ph.D. in mathematics from Oregon State University, Oregon, U.S.A. He is a professor of mathematics at the National University of Ireland, Galway. He is the author/co-author of 14 books and has published over 650 papers on fixed point theory, operator, integral, differential and difference equations. He serves on the editorial board of many mathematical journals.

Previously, the authors have co-authored/co-edited the following books with Springer: Infinite Interval Problems for Differential, Difference and Integral Equations; Singular Differential and Integral Equations with Applications; Nonlinear Analysis and Applications: To V. Lakshmikanthan on his 80th Birthday. In addition, they have collaborated with others on the following titles: Positive Solutions of Differential, Difference and Integral Equations; Oscillation Theory for Difference and Functional Differential Equations; Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations.

Table of contents :
Front Matter….Pages 1-12
Introduction….Pages 1-6
Historical Notes….Pages 7-12
Exact Equations….Pages 13-20
Elementary First-Order Equations….Pages 21-27
First-Order Linear Equations….Pages 28-34
Second-Order Linera Equations….Pages 35-44
Preliminaries to Existence and Uniqueness of Solutions….Pages 45-52
Picard’s Method of Successive Approximations….Pages 53-60
Existence Theorems….Pages 61-67
Uniqueness Theorems….Pages 68-76
Differential Inequalities….Pages 77-83
Continuous Dependence on Initial Conditions….Pages 84-90
Preliminary Results from Algebra and Analysis….Pages 91-96
Preliminary Results from Algebra and Analysis (Contd.)….Pages 97-102
Existence and Uniqueness of Solutions of Systems….Pages 103-108
Existence and Uniqueness of Solutions of Systems (Contd.)….Pages 109-115
General Properties of Linear Systems….Pages 116-123
Fundamental Matrix Solution….Pages 124-132
Systems with Constant Coefficients….Pages 133-143
Periodic Linear Systems….Pages 144-151
Asymptotic Behavior of Solutions of Linear Systems….Pages 152-158
Asymptotic Behavior of Solutions of Linear Systems (Contd.)….Pages 159-167
Preliminaries to Stability of Solutions….Pages 168-174
Stability of Quasi-Linear Systems….Pages 175-180
Two-Dimensional Autonomous Systems….Pages 181-186
Two-Dimensional Autonomous Systems (Contd.)….Pages 187-195
Limit Cycles and Periodic Solutions….Pages 196-203
Lyapunov’s Direct Method for Autonomous Systems….Pages 204-210
Lyapunov’s Direct Method for Nonautonomous Systems….Pages 211-216
Higher-Order Exact and Adjoint Equations….Pages 217-224
Oscillatory Equations….Pages 225-232
Linear Boundary Value Problems….Pages 233-239
Green’s Functions….Pages 240-249
Degenerate Linear Boundary Value Problems….Pages 250-257
Maximum Principles….Pages 258-264
Sturm—Liouville Problems….Pages 265-270
Sturm–Liouville Problems (Contd.)….Pages 271-278
Eigenfunction Expansions….Pages 279-285
Eigenfunction Expansions (Contd.)….Pages 286-294
Nonlinear Boundary Value Problems….Pages 295-299
Nonlinear Boundary Value Problems (Contd.)….Pages 300-307
Topics for Further Studies….Pages 308-314
Back Matter….Pages 1-12