Bernfeld S.R., Lakshmikantham V. An Introduction to Nonlinear Boundary Value Problems

Academic Press, 1974. — 386 p.The theory of nonlinear boundary value problems is an extremely important and interesting area of research in differential equations. Due to the entirely different nature of the underlying physical processes, its study is substantially more difficult than that of initial value problems and consequently belongs to a third course in differential equations. Although this sophisticated branch of research has, in recent years, developed significantly, the available books are either more elementary in nature, for example the book by Baily, Shampine, and Waltman, or directed to a particular method of importance, such as that by Bellman and Kalaba. Hence it is felt that a book on an advanced level that exposes the reader to this fascinating field of differential equations and provides a ready access to an up-to-date state of this art is of immense value. With this as motivation, we present in our book a variety of techniques that are employed in the theory of nonlinear boundary value problems.For example, we discuss the following:
methods that involve differential inequalities;
shooting and angular function techniques;
functional analytic approaches;
topological methods.
We have also included a chapter on nonlinear boundary value problems for functional differential equations and a chapter covering special topics of interest.The main features of the book are
a coverage of a portion of the material from the contribution of Russian mathematics of which the English speaking world is not well aware;
the use of several Lyapunov like functions and differential inequalities in a fruitful way;
the inclusion of many examples and problems to help the reader develop an expertise in the field.