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Theory of impulsive differential equations. (English) Zbl 0719.34002
Series in Modern Applied Mathematics, 6. Singapore etc.: World Scientific. x, 273 p. $ 58.00/hbk (1989).
The monograph summarizes results obtained by the authors and their colleagues and by the Soviet groups of Myshkis, Samoilenko, Perestyuk (there are 76 references, 57 belonging to these groups). Although there are illustrative examples (some of them interesting) there are no applications. With an extended use of inequalities and comparison arguments, most of the book is devoted to stability.
Chapter 1 introduces the impulsive evolution processes, presents preliminary results and examples. In Chapter 2 variation of parameters formulae, upper and lower solutions, monotone iterative techniques, simple stability criteria are considered. Chapter 3 is devoted to the study of stability by means of discontinuous Lyapunov functions and impulsive differential inequalities. In Chapter 4 different other aspects of impulsive systems are discussed.

MSC:
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A37 Ordinary differential equations with impulses
34A40 Differential inequalities involving functions of a single real variable
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34K05 General theory of functional-differential equations
34K10 Boundary value problems for functional-differential equations
34K20 Stability theory of functional-differential equations
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