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 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