Oscillation theory of impulsive differential equations.

*(English)*Zbl 0949.34002
Orlando, FL: International Publications, ii, 284 p. (1998).

From the preface: “Oscillation theory is one of the directions which initiated the investigation of the qualitative properties of differential equations. This theory started with the classical works of Sturm and Kneser, and still attracts the attention of many mathematicians as much for the interesting results obtained as for their various applications.

The attraction of oscillation theory is linked rather strongly with the occurrence of new objects to be investigated. Such fast development can be observed in studying the oscillation properties of differential equations with deviating argument after 1970, as well as in the investigations in this direction for neutral equations after 1980. The main part of these investigations has been presented in monographs of G. S. Ladde, V. Lakshmikantham and B. G. Zhang [Oscillation theory of differential equations with deviating arguments. Marcel Dekker. 110. New York, NY (1987; Zbl 0622.34071 and Zbl 0832.34071)], I. Győri and G. Ladas [Oscillation theory of delay differential equations: with applications. Oxford: Clarendon Press (1991; Zbl 0780.34048)], D. D. Bainov and D. P. Mishev [Oscillation theory for neutral differential equations with delay. Bristol etc.: Adam Hilger (1991; Zbl 0747.34037)] and L. H. Erbe, Q. Kong and B. G. Zhang [Oscillation theory for functional differential equations. New York: Marcel Dekker (1994; Zbl 0821.34067)].

The last decade has seen an intensive investigation of impulsive differential equations. Numerous aspects of the qualitative theory have been studied in the monographs of A. M. Samoilenko and N. A. Perestyuk (1987), Bainov, Lakshmikantham and Simeonov (1989) and D. D. Bainov and P. S. Simeonov [Systems with impulsive effect stability, theory and applications. New York etc.: Halsted Press (1989; Zbl 0683.34032); Impulsive differential equations: periodic solutions and applications. New York, NY: John Wiley & Sons, Inc. (1993; Zbl 0815.34001) and Impulsive differential equations. Asymptotic properties of the solutions. Singapore: World Scientific (1995; Zbl 0828.34002)].

In 1989 the paper of K. Gopalsamy and B. G. Zhang [J. Math. Anal. Appl. 139, No. 1, 110-122 (1989; Zbl 0687.34065)] was published, where the first investigation on oscillatory properties of impulsive differential equations was carried out.

During the last two years Bainov, M. B. Dimitrova, Yu. I. Domshlak, E. I. Minchev and Simeonov have studied the oscillatory properties of various classes of impulsive differential equations. The results obtained by Bainov and Simeonov form the basis of the present book.

The authors’ aim is to present systematically the results known up to now, and to demonstrate how well-known mathematical techniques and methods, after suitable modification, can be applied in proving oscillation theorems for impulsive differential equations.

The book is divided into nine chapters. Chapter I includes definitions, formulas and assertions used throughout the book. In Chapter II oscillation properties are considered for first- and second-order linear impulsive differential equations without deviation of the argument. The main attention is devoted to the Sturm comparison theory for linear impulsive differential equations of second order. Chapter III deals with oscillation properties of linear impulsive differential equations with deviating argument and variable coefficients. In Chapters IV and V the notions of characteristic systems and generalized characteristic systems are introduced. They are used later in obtaining various criteria for the existence of a positive solution as well as for the oscillation of solutions to linear impulsive differential equations of first order with one or several deviating arguments. Special attention is devoted to the periodic equations of this type. Chapter VI concerns oscillation properties of solutions to neutral impulsive differential equations of first order with constant coefficients. In Chapters VII and VIII oscillation properties are studied for some classes of nonlinear impulsive differential equations of first and second order. Finally, Chapter IX deals with oscillation properties of impulsive partial differential equations”.

The attraction of oscillation theory is linked rather strongly with the occurrence of new objects to be investigated. Such fast development can be observed in studying the oscillation properties of differential equations with deviating argument after 1970, as well as in the investigations in this direction for neutral equations after 1980. The main part of these investigations has been presented in monographs of G. S. Ladde, V. Lakshmikantham and B. G. Zhang [Oscillation theory of differential equations with deviating arguments. Marcel Dekker. 110. New York, NY (1987; Zbl 0622.34071 and Zbl 0832.34071)], I. Győri and G. Ladas [Oscillation theory of delay differential equations: with applications. Oxford: Clarendon Press (1991; Zbl 0780.34048)], D. D. Bainov and D. P. Mishev [Oscillation theory for neutral differential equations with delay. Bristol etc.: Adam Hilger (1991; Zbl 0747.34037)] and L. H. Erbe, Q. Kong and B. G. Zhang [Oscillation theory for functional differential equations. New York: Marcel Dekker (1994; Zbl 0821.34067)].

The last decade has seen an intensive investigation of impulsive differential equations. Numerous aspects of the qualitative theory have been studied in the monographs of A. M. Samoilenko and N. A. Perestyuk (1987), Bainov, Lakshmikantham and Simeonov (1989) and D. D. Bainov and P. S. Simeonov [Systems with impulsive effect stability, theory and applications. New York etc.: Halsted Press (1989; Zbl 0683.34032); Impulsive differential equations: periodic solutions and applications. New York, NY: John Wiley & Sons, Inc. (1993; Zbl 0815.34001) and Impulsive differential equations. Asymptotic properties of the solutions. Singapore: World Scientific (1995; Zbl 0828.34002)].

In 1989 the paper of K. Gopalsamy and B. G. Zhang [J. Math. Anal. Appl. 139, No. 1, 110-122 (1989; Zbl 0687.34065)] was published, where the first investigation on oscillatory properties of impulsive differential equations was carried out.

During the last two years Bainov, M. B. Dimitrova, Yu. I. Domshlak, E. I. Minchev and Simeonov have studied the oscillatory properties of various classes of impulsive differential equations. The results obtained by Bainov and Simeonov form the basis of the present book.

The authors’ aim is to present systematically the results known up to now, and to demonstrate how well-known mathematical techniques and methods, after suitable modification, can be applied in proving oscillation theorems for impulsive differential equations.

The book is divided into nine chapters. Chapter I includes definitions, formulas and assertions used throughout the book. In Chapter II oscillation properties are considered for first- and second-order linear impulsive differential equations without deviation of the argument. The main attention is devoted to the Sturm comparison theory for linear impulsive differential equations of second order. Chapter III deals with oscillation properties of linear impulsive differential equations with deviating argument and variable coefficients. In Chapters IV and V the notions of characteristic systems and generalized characteristic systems are introduced. They are used later in obtaining various criteria for the existence of a positive solution as well as for the oscillation of solutions to linear impulsive differential equations of first order with one or several deviating arguments. Special attention is devoted to the periodic equations of this type. Chapter VI concerns oscillation properties of solutions to neutral impulsive differential equations of first order with constant coefficients. In Chapters VII and VIII oscillation properties are studied for some classes of nonlinear impulsive differential equations of first and second order. Finally, Chapter IX deals with oscillation properties of impulsive partial differential equations”.

##### MSC:

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

34A37 | Ordinary differential equations with impulses |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35R10 | Functional partial differential equations |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

34K11 | Oscillation theory of functional-differential equations |