Existence Theory for Nonlinear Ordinary Differential Equations

Description

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.

Get the book

Amazon.com

Search the book

eBay.com

Search the book

Other editions

• 

  Existence Theory for Nonlinear Ordinary Differential Equations

  • 2013-04-17
  • 207 pages
  • Ebook
  • Springer Science & Business Media

Similar books

• 

  Existence Theory for Nonlinear Integral and Integrodifferential Equations

  By Donal O'Regan, Maria Meehan

  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 ...

• 

  Nonlinear Differential Equations and Dynamical Systems: Theory and Applications

  By Feliz Manuel Minhós, João Fialho

  This Special Edition contains new results on Differential and Integral Equations and Systems, covering higher-order Initial and Boundary Value Problems, fractional differential and integral equations and applications, non-local optimal ...

• 

  Nonlinear Differential Equations in Abstract Spaces

  By V. Lakshmikantham, S. Leela

  Many problems in partial differential equations which arise from physical models can be considered as ordinary differential equations in appropriate infinite dimensional spaces, for which elegant theories and powerful techniques...

• 

  Nonlinear Differential Equations in Ordered Spaces

  By Seppo Heikkilä, S. Carl

  Nonlinear Differential Equations in Ordered Spaces includes research that appears for the first time in book form and is designed as a source book for pure and applied mathematicians.

• 

  Lectures on Nonlinear Hyperbolic Differential Equations

  By Lars Hörmander

  Lax , P.D. [ 3 ] , Weak solutions of nonlinear hyperbolic equations in two independent variables , Comm . ... A note on the life - span of classical solutions to nonlinear wave equations in four space dimensions , Indiana Univ . Math .

• 

  Nonlinear Interpolation and Boundary Value Problems

  By Johnny Henderson, Paul W. Eloe

  "This book is devoted to the study of solutions of nonlinear ODE boundary value problems as nonlinear interpolation problems.

• 

  Solvability of Nonlinear Singular Problems for Ordinary Differential Equations

  By Irena Rachunkova, Svatoslav Stanek, Milan Tvrdy

  Solvability of Nonlinear Singular Problems for Ordinary Differential Equations

• 

  Playing Around Resonance: An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations

  By Alessandro Fonda

  This book provides an up-to-date description of the methods needed to face the existence of solutions to some nonlinear boundary value problems.

• 

  Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition

  By Jane Cronin

  This third edition of a highly acclaimed textbook provides a detailed account of the Bendixson theory of solutions of two-dimensional nonlinear autonomous equations, which is a classical subject that has become more prominent in recent ...

• 

  Differential Equations: Theory and Current Research

  By Terry E. Moschandreou

  The book is written primarily for those who have some knowledge of differential equations and mathematical analysis. The authors of each section bring a strong emphasis on theoretical foundations to the book.