Ordinary differential equations. An introduction to nonlinear analysis. Transl. from the German by Gerhard Metzen
The book under review consists of 6 chapters with 27 sections of nearly equal size. The first chapter gives an extended introduction to various fields where ordinary differential equations occur, and also sketches some classes of equations which admit solutions in closed form. As a matter of fact, most equations cannot be solved explicitly (or, stated more drastically, all equations which may be solved have been solved). This emphasizes the need of providing information on the qualitative behaviour of hypothetic solutions, without knowing them, and this is the main purpose of the Chapters 2-6 of this book. The contents of these chapters go as follows. Chapter 2 is concerned with the usual existence and continuity theorems for initial value problems in Euclidean space. The last section of this chapter contains elementary results on semiflows and flows. The theory of linear equations is treated in detail in Chapter 3. As usual, particular emphasis is put on properties of the Cauchy function (evolution operator) generated by a strongly continuous linear operator function \(A=A(t)\). As is well known, the stationary case A(t)\(\equiv A\) admits a complete and precise description, e.g. in the classification of equilibrium points, for necessary and sufficient stability conditions, or in characterizing the \(\epsilon\)-dichotomy (hyperbolicity) of critical points. Qualitative properties are dealt with in Chapter 4, pointing out the importance of Lyapunov stability and Lyapunov functions, limit sets and attractors, stable and unstable manifolds. Moreover, the Grobman-Hartman linearization principle (at hyperbolic critical points) is discussed, and some consequences are derived. Chapter 5 is entirely devoted to periodic solutions. The simple observation that the fixed points of the Poincaré operator (translation operator along the trajectories) are precisely the initial values which give rise to periodic solutions, motivates the need of treating Brouwer's fixed point principle, which is done here by means of topological degree theory. Apart from existence, stability of periodic solutions is discussed in detail, building on Floquet theory, Lyapunov functions, and the Poincaré recurrence theorem. The Poincaré-Bendixson theorem for planar flows is given in a separate section. At this point, one should mention that the subtitle of the book is quite misleading. Of course, this book is not an introduction to nonlinear analysis, but simply employs, among many other methods, also some auxiliary tools and techniques from nonlinear analysis. This refers mainly to the degree and fixed point methods mentioned above, and to general results in bifurcation theory developed in Chapter 6. Here the Lyapunov-Schmidt reduction is, of course, of central interest. Hopf bifurcation and various ``interactions'' between stability and bifurcation are dealt with in the remaining part of this chapter. The book contains a wealth of useful information and is extremely carefully written. The amount of misprints and oversights (e.g., Problem 3 on p. 368) is surprisingly small. The systematic structure of the definitions, propositions, and remarks, as well as a concise summary at the beginning of each chapter, provide a good orientation for the reader. Several exercises, most of them being elementary, are added to each of the 27 sections. The reviewer did not like at all, however, the first chapter. Here the author collects, in a rather confusing way, many topics and notions which are completely unnecessary for understanding or even motivating the rest of the book, and which definitely cannot be understood by ``students in their second year of studies'' (as claimed in the Preface). Accordingly, the exercises in this chapter (``prove Remark xxx'', ``show that Proposition xxx is true'', etc.) are less interesting than in the other chapters. Apart from this flaw, this is one of the best books on ordinary differential equations the reviewer is aware of.
scientific article; zbMATH DE number 194474