Existence Theory for Nonlinear Ordinary Differential Equations

Donal O'Regan

ISBN 10: 9048148359 / ISBN 13: 9789048148356

Verlag: Springer Netherlands, 2010

Beschreibung

Druck auf Anfrage Neuware - Printed after ordering - 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.

Bestandsnummer des Verkäufers 9789048148356

Bibliografische Details

• Titel: Existence Theory for Nonlinear Ordinary Differential Equations
• Verlag: Springer Netherlands
• Erscheinungsjahr: 2010
• Sprache: Englisch
• ISBN-10: 9048148359
• ISBN-13: 9789048148356
• Einband: Taschenbuch
• Zustand: Neu
• Abmessungen: 235B x 155 H x 12 T Millimeter Breite: 235 mm, Höhe: 155 mm, Tiefe: 12 mm
• Gewicht: 330 Gramm

Über diesen Titel

Inhaltsangabe

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.

Críticas

`I warmly recommend it to everyone interested in ordinary differential equations and its applications.'
Acta Sci. Math., 64 (1998)