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Trends in the theory of impulsive differential equations. (English) Zbl 0718.34011

Differential equations and applications, Proc. Int. Conf., Columbus/OH (USA) 1988, Vol. II, 76-87 (1989).
[For the entire collection see Zbl 0707.00015.]
It is now realized that the theory of impulsive differential equations is an important area of investigation and its theory is a lot richer than the corresponding theory of differential equations. Moreover, such equations seem to represent a natural framework for mathematical modelling of several real world phenomena. For example, if the population of a given species is regulated by some impulse factors at certain moments, then it is not reasonable to expect a regular solution. Instead, the solutions must have some jumps at these moments, and the jumps, of course, follow some specific pattern. Furthermore, the reproduction of microorganisms suggests that periodic solutions exist under certain impulsive conditions. This is indeed the motivation for the present paper in which we discuss the recent trends in the theory of impulsive differential equations. Since much of the work is initiated in USSR of which the English speaking readers are not well aware of, we describe some important questions highlighting the known theory and open problems.

MSC:

34A37 Ordinary differential equations with impulses
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 0707.00015