Principles of Differential and Integral Equations

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.