Generalized differential equations. Fundamental results. (English) Zbl 0594.34002

Schwabik, Štefan

Rozpr. Česk. Akad. Věd, Řada Mat. Přír. Věd 95, No. 6, 103 p. (1985).

This publication has the character of a monograph and is devoted to the theory of generalized ordinary differential equations. This theory was created in 1956 by Jaroslav Kurzweil and was elaborated by many authors. This paper is the first synthetic work on this field. The basis of the theory of generalized differential equations consists in the theory of the generalized Perron integral defined by J. Kurzweil. Fundamental existence and uniqueness theorems are given. The properties of the set of solutions are studied (Kneser-type theorem), great attention is devoted to the systems of linear equations, particularly with the periodic coefficients (the generalized version of the Floquet theory). The relation between the generalization and the classical theory is studied in detail. By means of the generalized differential equations a lot of problems of the theory of ordinary differential equations may be studied. The special problems will be continued in another publication.

Reviewer: M.Bartušek

MSC:

34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A30	Linear ordinary differential equations and systems

Keywords:

generalized Perron integral; Kneser-type theorem; periodic coefficients; Floquet theory