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Existence Theory for Nonlinear Ordinary Differential Equations

  • Donal O’Regan

Part of the Mathematics and Its Applications book series (MAIA, volume 398)

Table of contents

  1. Front Matter
    Pages i-vii
  2. Donal O’Regan
    Pages 1-3
  3. Donal O’Regan
    Pages 4-10
  4. Donal O’Regan
    Pages 11-21
  5. Donal O’Regan
    Pages 22-33
  6. Donal O’Regan
    Pages 58-66
  7. Donal O’Regan
    Pages 67-81
  8. Donal O’Regan
    Pages 116-132
  9. Donal O’Regan
    Pages 133-155
  10. Donal O’Regan
    Pages 156-163
  11. Donal O’Regan
    Pages 174-185
  12. Donal O’Regan
    Pages 186-194
  13. Back Matter
    Pages 195-200

About this book

Introduction

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.

Keywords

Boundary value problem differential equation ordinary differential equation

Authors and affiliations

  • Donal O’Regan
    • 1
  1. 1.Department of MathematicsUniversity College GalwayGalwayIreland

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-1517-1
  • Copyright Information Springer Science+Business Media B.V. 1997
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4835-6
  • Online ISBN 978-94-017-1517-1
  • Buy this book on publisher's site
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