Basic theory of ordinary differential equations. (English) Zbl 0924.34001

Hsieh, Po-Fang; Sibuya, Yasutaka

Universitext. New York, NY: Springer. xi, 468 p. (1999).

This graduate level textbook consists of thirteen chapters the particular contents of which are as follows:
Fundamental existence and uniqueness theorems and smoothness in data of an initial value problem are treated in Chapters I and II, whereas the results concerning nonuniqueness are discussed in Chapter III. Chapter IV presents basic theorems of linear differential equations. In particular, systems with constant or periodic coefficients are treated in detail. The authors show the importance of the well-known \(S-N\) decomposition theorem saying that every matrix \(A\in M_n (\mathbb{C})\) can be uniquely presented in the form \(A=S+N\) with \(S\) diagonalizable and \(N\) nilpotent and satisfying \(SN=NS\). The method based on this fact is applied in subsequent chapters of this book.
In Chapter V is introduced the notion of formal power series \(\sum^\infty_{n=0} a_nx^n\) the totality of which constitutes, in a natural way, a differentiable algebra \(\mathbb{C}[[x]]\). Adopting this algebraic point of view, the authors treat the differential equation \(F(x,y,{dy\over dx},\dots, {d^ny\over dx^n})=0\) where \(F\) is a polynomial in its variables. Chapters VI and X deal with Sturm-Liouville problems. Asymptotic behavior of solutions and stability of systems are treated in Chapters VII and VIII. Autonomous systems and the Poincaré-Bendixson theorem are discussed in Chapter IX. Finally, asymptotic expansions and singularities are discussed in Chapters XI, XII and XIII.

Reviewer: L.Janos (Kent/Ohio)

MSC:

34-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
34B24	Sturm-Liouville theory
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A30	Linear ordinary differential equations and systems
34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34D05	Asymptotic properties of solutions to ordinary differential equations
34D20	Stability of solutions to ordinary differential equations

Keywords:

asymptotic behavior; existence; uniqueness; linear differential equations; periodic coefficients; decomposition theorem; formal power series; Sturm-Liouville problems; solutions; stability; Poincaré-Bendixson theorem; singularities