From the Preface: 'The aim of this book is to offer a discussion of the fundamental concepts in the theory of differential and integral equations, with a view toward preparing the reader for graduate-level courses on advanced topics. This book can also serve a graduate-level course for non-mathematics majors. Students of various disciplines have found an increasing need to study such topics as control theory and systems theory, which are mainly based on differential equations, integral operators, and related concepts. The material included in this text will provide sufficient basis for understanding these areas of investigation. The prerequisites are a good course in advanced calculus...some preparation in linear algebra, and a reasonable acquaintance with elementary complex analysis...The book can be used as a first course in differential and integral equations by any reader possessing knowledge of the above-mentioned fields'.
Introduction: Some Elementary Methods: The linear equation of the first order The equation with separable variables Exact differential equations The uniqueness problem: an example Some integral inequalities Problems Existence Theory for Differential Equations: The first-order equation First-order differential systems Equations and systems of higher order Peano existence theorem A uniqueness theorem Problems Some Global Problems for Ordinary Differential Equations: Statement of the problems Global uniqueness Global existence and the behavior of saturated solutions Dependence of solutions on initial values Differential inequalities and the comparison method A criterion of global existence Problems Some Special Classes of Differential Systems and Equations: Linear systems: generalities Linear homogeneous systems Nonhomogeneous systems Linear equations of higher order Autonomous systems Linear systems and equations with constant coefficients Problems Stability Theory of Ordinary Differential Systems: Definitions and examples Stability of linear systems Stability in the first approximation Stability theorems by comparison method Further stability results Stability of automatic control systems Problems Volterra Integral Equations: Existence and uniqueness of solution An existence theorem The linear equation The first-kind linear equation Some problems on the half-axis Problems Fredholm Theory of Linear Integral Equations: The resolvent kernel The entire functions of Fredholm and their applications A glimpse of Hilbert space theory Eigenvalues, eigenfunctions and adjoint equations Problems Theory of Self-Adjoint Integral Equations and Some Applications: Some properties of an integral operator The existence of eigenvalues The Hilbert-Schmidt expansion theorem Complete kernels and systems The Sturm-Liouville problem Reduction to an integral equation: the first case Reduction to an integral equation: the second case Problems Miscellanea (Nice Things): Maximum and minimum solutions of scalar differential equations A theorem of Massera An oscillation theorem A formula of Picone and its consequences Lienard's equation Differential systems without uniqueness Abel's integral equation References Index.
