Impulsive differential equations. Asymptotic properties of the solutions.

*(English)*Zbl 0828.34002
Series on Advances in Mathematics for Applied Sciences. 28. Singapore: World Scientific. x, 230 p. (1995).

This work is the fourth monograph on the impulsive theory written by D. Bainov and P. Simeonov during the last 6 years. In this book, the authors study the asymptotic behaviour as \(t\to +\infty\) of solutions to linear and weakly nonlinear impulsive systems of the form (1) \(dx/dt= A(t) x+ f(t, x)\), (2) \(\Delta x|_{t_i}= B_i x+ h_i(x)\), where \(x\in \mathbb{R}^n\), \(t_i\in \mathbb{R}_+\), \(\lim_{i\to \infty} t_i= +\infty\), \(\text{det}(E+ B_i)\neq 0 \forall i\).

The main aim of the investigation is to establish the asymptotic relations similar to \(x(t)= (a+ o(1)) y(t)\), \(t\to +\infty\) for solutions \(x(t)\) of various classes of (1), (2). Here \(a\in \mathbb{R}^n\) and \(y(t)\) is a solution of some comparison impulsive system (frequently this comparison system is integrable). Note that the analogous problem for the equation (1) was studied previously by N. Levinson, P. Hartman and A. Wintner, M. S. P. Eastham, M. Pinto and other authors, and the present book extends mostly their results to the impulsive case. It is worth pointing out that the current topic is not covered by any other monograph devoted to the impulsive theory including the pioneer work [SP] of A. M. Samojlenko and N. A. Perestyuk [Impulsive differential equations. (English). World scientific series on mathematical science. Series A, Vol. 14. Singapore: World Scientific, 462 p. (1995)] (however, see [SP] to complete the references).

All of the problems in this book are investigated in an elementary way by means of the classical apparatus of ordinary differential equations and mathematical analysis. The text can be used by graduate and Ph.D. students as well as by researchers in the field of discontinuous dynamical systems and their applications.

Contents: Preliminary notes; Asymptotic formulae; Convergence of the solutions; Asymptotic equivalence.

The main aim of the investigation is to establish the asymptotic relations similar to \(x(t)= (a+ o(1)) y(t)\), \(t\to +\infty\) for solutions \(x(t)\) of various classes of (1), (2). Here \(a\in \mathbb{R}^n\) and \(y(t)\) is a solution of some comparison impulsive system (frequently this comparison system is integrable). Note that the analogous problem for the equation (1) was studied previously by N. Levinson, P. Hartman and A. Wintner, M. S. P. Eastham, M. Pinto and other authors, and the present book extends mostly their results to the impulsive case. It is worth pointing out that the current topic is not covered by any other monograph devoted to the impulsive theory including the pioneer work [SP] of A. M. Samojlenko and N. A. Perestyuk [Impulsive differential equations. (English). World scientific series on mathematical science. Series A, Vol. 14. Singapore: World Scientific, 462 p. (1995)] (however, see [SP] to complete the references).

All of the problems in this book are investigated in an elementary way by means of the classical apparatus of ordinary differential equations and mathematical analysis. The text can be used by graduate and Ph.D. students as well as by researchers in the field of discontinuous dynamical systems and their applications.

Contents: Preliminary notes; Asymptotic formulae; Convergence of the solutions; Asymptotic equivalence.

Reviewer: S.I.Trofimchuk (Kiev)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34A37 | Ordinary differential equations with impulses |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34D10 | Perturbations of ordinary differential equations |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |