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