Series:

Mathematics and Its Applications

2 primary works

Book 398

Cover of Existence Theory for Nonlinear Ordinary Differential Equations
We begin our applications of fixed point methods with existence of solutions to certain first order initial initial value problems. This problem is relatively easy to treat, illustrates important methods, and in the end will carry us a good deal further than may first meet the eye. Thus, we seek solutions to Y'. = I(t,y) (1. 1 ) { yeO) = r n where I: I X R n ---+ R and I = [0, b]. We shall seek solutions that are de fined either locally or globally on I, according to the assumptions imposed on I. Notice that (1. 1) is a system of first order equations because I takes its values in Rn. In section 3. 2 we will first establish some basic existence theorems which guarantee that a solution to (1. 1) exists for t > 0 and near zero. Familiar examples show that the interval of existence can be arbi trarily short, depending on the initial value r and the nonlinear behaviour of I. As a result we will also examine in section 3. 2 the dependence of the interval of existence on I and r. We mention in passing that, in the results which follow, the interval I can be replaced by any bounded interval and the initial value can be specified at any point in I. The reasoning needed to cover this slightly more general situation requires minor modifications on the arguments given here.

Book 445

Cover of Existence Theory for Nonlinear Integral and Integrodifferential Equations
The theory of integral and integrodifferential equations has ad­ vanced rapidly over the last twenty years. Of course the question of existence is an age-old problem of major importance. This mono­ graph is a collection of some of the most advanced results to date in this field. The book is organized as follows. It is divided into twelve chap­ ters. Each chapter surveys a major area of research. Specifically, some of the areas considered are Fredholm and Volterra integral and integrodifferential equations, resonant and nonresonant problems, in­ tegral inclusions, stochastic equations and periodic problems. We note that the selected topics reflect the particular interests of the authors. Donal 0 'Regan Maria Meehan CHAPTER 1 INTRODUCTION AND PRELIMINARIES 1.1. Introduction The aim of this book is firstly to provide a comprehensive existence the­ ory for integral and integrodifferential equations, and secondly to present some specialised topics in integral equations which we hope will inspire fur­ ther research in the area. To this end, the first part of the book deals with existence principles and results for nonlinear, Fredholm and Volterra inte­ gral and integrodifferential equations on compact and half-open intervals, while selected topics (which reflect the particular interests of the authors) such as nonresonance and resonance problems, equations in Banach spaces, inclusions, and stochastic equations are presented in the latter part.