Differential equations with discontinuous right-hand sides. Ed. by F. M. Arscott. Transl. from the Russian. (English) Zbl 0664.34001

Filippov, A. F.

Mathematics and Its Applications: Soviet Series, 18. Dordrecht etc.: Kluwer Academic Publishers. x, 304 p. £59.00; $ 99.00; Dfl. 198.00 (1988).

The book deals with the ordinary differential equations and systems of such equations whose right-hand sides (or coefficients) are discontinuous functions of x or/and t. Many results are presented in this book which were earlier unavailable in book form. The book consists of a very brief introduction, where the concept of the solution is discussed, and 5 chapters: 1. Equations with the right-hand side continuous in x and discontinuous in t. 2. Existence and general properties of solutions of discontinuous systems. 3. Basic methods of qualitative theory. 4. Local singularities of two-dimensional systems. 5. Local-singularities of three-dimensional and multidimensional systems. The bibliography consists of 208 entries, most of them are papers in Russian. The subject index is given. The author treats Caratheodory differential equations, equations with distributional coefficients and right-hand sides, differential inclusions, dependence of the solutions on the initial data, properties of the trajectories, boundedness, periodicity and stability of the solutions, classification of singular points in two-, three-, and multidimensional systems. This monograph is a valuable contribution to the literature which is of interest to everybody who studies or uses ordinary differential equations.

Reviewer: A.Ramm

MSC:

34-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
34A60	Ordinary differential inclusions
34C25	Periodic solutions to ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20	Stability of solutions to ordinary differential equations

Keywords:

Local singularities of two-dimensional systems; multidimensional systems; Caratheodory differential equations; equations with distributional coefficients; differential inclusions; trajectories; boundedness; singular points