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Impulsive differential equations with a small parameter. (English) Zbl 0828.34001

Series on Advances in Mathematics for Applied Sciences. 24. Singapore: World Scientific. viii, 268 p. (1994).
The monograph under review deals with the impulsive differential equations (IDE) with a small parameter and consists of three chapters. The first one has an introductory character and contains examples of the problems (without impulses) involving a small parameter along with the basic information on IDE. The second chapter is devoted to the regularly perturbed IDE and presents mainly the results on the averaging and partially additive/multiplicative averaging of IDE in a standard form. The existence of periodic/almost periodic solutions of IDE and the existence of integral sets for IDE with hyperbolic linear parts are also discussed in this chapter. The principal topics considered in the third chapter are initial value problems, two-point boundary value problems and periodic problems for the singularly perturbed IDE. The investigation of these problems is carried out with the help of the method of boundary functions due to A. B. Vasil’eva and V. F. Butuzov, the numerical- analytical method due to A. M. Samoĭlenko and the averaging method due to N. N. Bogolyubov and Yu. A. Mitropol’skiĭ. Apart from the theorems presented in sections 2-4 and 9.1 adopted with slight modifications from the monograph of A. M. Samoĭlenko and N. A. Perestyuk [Differential equations with impulse action. Kiev: Vishcha Shkola (1987); English translation (1995; Zbl 0837.34003)] and that in section 7 adopted from the paper of A. M. Samoĭlenko [Mat. Fiz. Nelineĭn. Mekh. 9, 101–117 (1971; Zbl 0305.34067)] the book outlines the results on IDE with small parameter obtained during the last decade by Bulgarian mathematicians. It should be noted that the results of section 4 were improved in recent papers by A. M. Samoĭlenko and S. I. Trofimchuk [Differ. Equations 29, 684–691 (1993)], M. U. Akhmetov and N. A. Perestyuk [J. Appl. Math. Mech. 56, 829–837 (1992; Zbl 0791.34034) and Ukr. Math. J. 43, 1208-1214 (1991; Zbl 0786.34004)], while for the recent results related to that in section 9.1 we refer to the papers by M. U. Akhmetov and N. A. Perestyuk [Ukr. Math. J. 44, 1–7 (1992; Zbl 0786.34005)] and O. V. Vyshenskaya and N. A. Perestyuk [Ukr. Math. J. 44, 528–534 (1992; Zbl 0782.34053)].
In order to get a more complete picture of the subject the prospective reader should also consult a great number of papers on different important classes of the problems for IDE with a small parameter which are not even mentioned in the monograph. Being far from the intension to give an exhausting bibliography for all results which are not covered by the authors of the book we just refer to some papers which in our opinion are characteristic for the corresponding directions of the research. Namely, regarding the partial differential equations with impulses we note that the hyperbolic impulsive systems with a small parameter were studied by the reviewer [Ukr. Math. J. 40, 212–215 (1988; Zbl 0681.35059)] and by Yu. A. Mitropol’skiĭ and the reviewer [Mat. Fiz. Nelineĭn. Mekh. 47, No. 13, 61–67 (1990)], while the impulse evolution system defined by an abstract parabolic equation was considered by Yu. A. Mitropol’skiĭ and Yu. V. Rogovchenko [Ukr. Math. J. 44, 76- -83 (1992; Zbl 0786.34048)].
For the study of multifrequency oscillatory IDE we refer to the papers by the reviewer [Asymptotic methods in problems of mathematical physics, Collect. Sci. Works, Kiev, 105–111 (1989; Zbl 0721.34046)] and by A. M. Samojlenko and M. N. Astaf’eva [Dokl. Akad. Nauk Ukrain. SSR, Ser. A, 12–16 (1989; Zbl 0687.34032)]. It is also impossible to avoid citing the survey by Yu. A. Mitropol’skiĭ, A. M. Samoĭlenko and N. A. Perestyuk on the applications of asymptotic methods for the study of IDE with a small parameter [Ukr. Math. J. 37, 48–55 (1985; Zbl 0583.34036)]. Finally, there are inaccuracies in the text which may create some difficulties for the reader (for example, equation (1.2) on p. 1 should be written as \(\dot{\overline x}= f(t, \overline x, 0)\), the definition of a solution to the IDE (2.4) on p. 10 taken from the cited above monograph by Samoĭlenko and Perestyuk should be corrected as follows: after the words ‘for all \(t\neq t_i\)’ it is necessary to insert ‘satisfying the condition of the jump at the points \(t= t_i\)’, etc.).

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A37 Ordinary differential equations with impulses
34C25 Periodic solutions to ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34C29 Averaging method for ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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