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..

Stuart P. Hastings, *University of Pittsburgh, PA*, and J. Bryce McLeod, *Oxford University, England, and University of Pittsburgh, PA*

*Preface*

**Chapter 1. Introduction**

1.1. What are classical methods?

1.2. Exercises

**Chapter 2. An introduction to shooting methods**

2.1. Introduction

2.2. A first order example

2.3. Some second order examples

2.4. Heteroclinic orbits and the FitzHugh-Nagumo equations

2.5. Shooting when there are oscillations: A third order problem

2.6. Boundedness on (-8,8) and two-parameter shooting

2.7. Waz?ewski's principle, Conley index, and an n-dimensional lemma

2.8. Exercises

**Chapter 3. Some boundary value problems for the Painlev´e transcendents**

3.1. Introduction

3.2. A boundary value problem for Painlev´e

3.3. Painlev´e II—shooting from infinity

3.4. Some interesting consequences

3.5. Exercises

**Chapter 4. Periodic solutions of a higher order system**

4.1. Introduction, Hopf bifurcation approach

4.2. A global approach via the Brouwer fixed point theorem

4.3. Subsequent developments

4.4. Exercises

**Chapter 5. A linear example**

5.1. Statement of the problem and a basic lemma

5.2. Uniqueness

5.3. Existence using Schauder's fixed point theorem

5.4. Existence using a continuation method

5.5. Existence using linear algebra and finite dimensional continuation

5.6. A fourth proof

5.7. Exercises

**Chapter 6. Homoclinic orbits of the FitzHugh-Nagumo equations**

6.1. Introduction

6.2. Existence of two bounded solutions

6.3. Existence of homoclinic orbits using geometric perturbation theory

6.4. Existence of homoclinic orbits by shooting

6.5. Advantages of the two methods

6.6. Exercises

**Chapter 7. Singular perturbation problems—rigorous matching**

7.1. Introduction to the method of matched asymptotic expansions

7.2. A problem of Kaplun and Lagerstrom

7.3. A geometric approach

7.4. A classical approach

7.5. The case n = 3

7.6. The case n = 2

7.7. A second application of the method

7.8. A brief discussion of blow-up in two dimensions

7.9. Exercises

Chapter 8. Asymptotics beyond all orders

8.1. Introduction

8.2. Proof of nonexistence

8.3. Exercises

**Chapter 9. Some solutions of the Falkner-Skan equation**

9.1. Introduction

9.2. Periodic solutions

9.3. Further periodic and other oscillatory solutions

9.4. Exercises

**Chapter 10. Poiseuille flow: Perturbation and decay**

10.1. Introduction

10.2. Solutions for small data

10.3. Some details

10.4. A classical eigenvalue approach

10.5. On the spectrum of D?,R? for large R

10.6. Exercises

**Chapter 11. Bending of a tapered rod; variational methods and shooting**

11.1. Introduction

11.2. A calculus of variations approach in Hilbert space

11.3. Existence by shooting for p > 2

11.4. Proof using Nehari's method

11.5. More about the case p = 2

11.6. Exercises

**Chapter 12. Uniqueness and multiplicity**

12.1. Introduction

12.2. Uniqueness for a third order problem

12.3. A problem with exactly two solutions

12.4. A problem with exactly three solutions

12.5. The Gelfand and perturbed Gelfand equations in three dimensions

12.6. Uniqueness of the ground state for ?u - u + u3 = 0

12.7. Exercises

**Chapter 13. Shooting with more parameters**

13.1. A problem from the theory of compressible flow

13.2. A result of Y.-H. Wan

13.3. Exercise

13.4. Appendix: Proof of Wan's theorem

**Chapter 14. Some problems of A. C. Lazer**

14.1. Introduction

14.2. First Lazer-Leach problem

14.3. The pde result of Landesman and Lazer

14.4. Second Lazer-Leach problem

14.5. Second Landesman-Lazer problem

14.6. A problem of Littlewood, and the Moser twist technique

14.7. Exercises

**Chapter 15. Chaotic motion of a pendulum**

15.1. Introduction

15.2. Dynamical systems

15.3. Melnikov's method

15.4. Application to a forced pendulum

15.5. Proof of Theorem 15.3 when d = 0

15.6. Damped pendulum with nonperiodic forcing

15.7. Final remarks

15.8. Exercises

**Chapter 16. Layers and spikes in reaction-diffusion equations, I**

16.1. Introduction

16.2. A model of shallow water sloshing

16.3. Proofs

16.4. Complicated solutions ("chaos")

16.5. Other approaches

16.6. Exercises

**Chapter 17. Uniform expansions for a class of second order problems**

17.1. Introduction

17.2. Motivation

17.3. Asymptotic expansion

17.4. Exercise

**Chapter 18. Layers and spikes in reaction-diffusion equations, II**

18.1. A basic existence result

18.2. Variational approach to layers

18.3. Three different existence proofs for a single layer in asimple case

18.4. Uniqueness and stability of a single layer

18.5. Further stable and unstable solutions, including multiple layers

18.6. Single and multiple spikes

18.7. A different type of result for the layer model

18.8. Exercises

**Chapter 19. Three unsolved problems**

19.1. Homoclinic orbit for the equation of a suspension bridge

19.2. The nonlinear Schr¨odinger equation

19.3. Uniqueness of radial solutions for an elliptic problem

19.4. Comments on the suspension bridge problem

19.5. Comments on the nonlinear Schr¨odinger equation

19.6. Comments on the elliptic problem and a new existence proof

19.7. Exercises

Bibliography

*Index *