On Semicontinuity in Impulsive Dynamical Systems

Volume 52 / 2004

Krzysztof Ciesielski Bulletin Polish Acad. Sci. Math. 52 (2004), 71-80 MSC: Primary 37B99; Secondary 54H20, 34A37. DOI: 10.4064/ba52-1-8

Abstract

In the important paper on impulsive systems [K1] several notions are introduced and several properties of these systems are shown. In particular, the function $\phi $ which describes “the time of reaching impulse points” is considered; this function has many important applications. In [K1] the continuity of this function is investigated. However, contrary to the theorem stated there, the function $\phi $ need not be continuous under the assumptions given in the theorem. Suitable examples are shown in this paper. We characterize the function $\phi $ from the point of view of its semicontinuity. Also, we show the analogous properties for impulsive systems given by semidynamical systems. In the last section we investigate the continuity properties of the escape time function in impulsive systems.

Authors

  • Krzysztof CiesielskiMathematics Institute
    Jagiellonian University
    Reymonta 4
    30-059 Kraków, Poland
    e-mail

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