This text emphasizes rigorous mathematical techniques for the analysis of boundary value problems for ODEs arising in applications. The emphasis is on proving existence of solutions, but there is also a substantial chapter on uniqueness and multiplicity questions and several chapters which deal with the asymptotic behavior of solutions with respect to either the independent variable or some parameter. These equations may give special solutions of important PDEs, such as steady state or traveling wave solutions. Often two, or even three, approaches to the same problem are described. The advantages and disadvantages of different methods are discussed.

The book gives complete classical proofs, while also emphasizing the importance of modern methods, especially when extensions to infinite dimensional settings are needed. There are some new results as well as new and improved proofs of known theorems. The final chapter presents three unsolved problems which have received much attention over the years.

Both graduate students and more experienced researchers will be interested in the power of classical methods for problems which have also been studied with more abstract techniques. The presentation should be more accessible to mathematically inclined researchers from other areas of science and engineering than most graduate texts in mathematics.

Table of Contents

Preface Introduction
What are classical methods? Exercises

An introduction to shooting methods
Introduction
A first order example
Some second order examples
Heteroclinic orbits and the FitzHugh-Nagumo equations
Shooting when there are oscillations: A third order problem
Boundedness on (−∞,∞) and two-parameter shooting
Waz˙ewski’s principle, Conley index, and an n-dimensional lemma Exercises

Some boundary value problems for the Painlev´e transcendents Introduction
A boundary value problem for Painlev´e
Painlev´e II—shooting from infinity
Some interesting consequences Exercises

Periodic solutions of a higher order system
Introduction, Hopf bifurcation approach
A global approach via the Brouwer fixed point theorem
Subsequent developments Exercises

A linear example
Statement of the problem and a basic lemma
Uniqueness
Existence using Schauder’s fixed point theorem
Existence using a continuation method
Existence using linear algebra and finite dimensional continuation
A fourth proof Exercises

Homoclinic orbits of the FitzHugh-Nagumo equations
Introduction
Existence of two bounded solutions
Existence of homoclinic orbits using geometric perturbation theory
Existence of homoclinic orbits by shooting
Advantages of the two methods Exercises

Singular perturbation problems—rigorous matching
Introduction to the method of matched asymptotic expansions
A problem of Kaplun and Lagerstrom
A geometric approach
A classical approach
The case n = 3
The case n = 2
A second application of the method
A brief discussion of blow-up in two dimensions Exercises

Asymptotics beyond all orders
Introduction
Proof of nonexistence Exercises

Some solutions of the Falkner-Skan equation
Introduction
Periodic solutions
Further periodic and other oscillatory solutions Exercises

Poiseuille flow: Perturbation and decay
Introduction
Solutions for small data
Some details
A classical eigenvalue approach
On the spectrum of Dξ,Rξ for large R Exercises

Bending of a tapered rod; variational methods and shooting
Introduction
A calculus of variations approach in Hilbert space
Existence by shooting for p > 2
Proof using Nehari’s method
More about the case p = 2 Exercises

Uniqueness and multiplicity
Introduction
Uniqueness for a third order problem
A problem with exactly two solutions
A problem with exactly three solutions
The Gelfand and perturbed Gelfand equations in three dimensions
Uniqueness of the ground state for Δu − u + u3 = 0 Exercises

Shooting with more parameters
A problem from the theory of compressible flow
A result of Y.-H. Wan Exercise Appendix: Proof of Wan’s theorem

Some problems of A. C. Lazer
Introduction
First Lazer-Leach problem
The pde result of Landesman and Lazer
Second Lazer-Leach problem
Second Landesman-Lazer problem
A problem of Littlewood, and the Moser twist technique Exercises

Chaotic motion of a pendulum
Introduction
Dynamical systems
Melnikov’s method
Application to a forced pendulum
Proof of Theorem 15.3 when δ = 0
Damped pendulum with nonperiodic forcing
Final remarks Exercises

Layers and spikes in reaction-diffusion equations, I
Introduction
A model of shallow water sloshing
Proofs
Complicated solutions ("chaos")
Other approaches Exercises

Uniform expansions for a class of second order problems
Introduction
Motivation
Asymptotic expansion Exercise

Layers and spikes in reaction-diffusion equations, II
A basic existence result
Variational approach to layers
Three different existence proofs for a single layer in asimple case
Uniqueness and stability of a single layer
Further stable and unstable solutions, including multiple layers
Single and multiple spikes
A different type of result for the layer model Exercises

Three unsolved problems
Homoclinic orbit for the equation of a suspension bridge
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