We introduce the notions of Lyapunov quasi-stability and Zhukovskiĭ quasi-stability of a trajectory in an impulsive semidynamical system defined in a metric space, which are counterparts of corresponding stabilities in the theory of dynamical systems. We initiate the study of fundamental properties of those quasi-stable trajectories, in particular, the structures of their positive limit sets. In fact, we prove that if a trajectory is asymptotically Lyapunov quasi-stable, then its limit set consists of rest points, and if a trajectory in a locally compact space is uniformly asymptotically Zhukovskiĭ quasi-stable, then its limit set is a rest point or a periodic orbit. Also, we present examples to show the differences between variant quasi-stabilities. Further, some sufficient conditions are given to guarantee the quasi-stabilities of a prescribed trajectory.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-2-4,
author = {Changming Ding},
title = {Lyapunov quasi-stable trajectories},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {139-154},
zbl = {1278.37024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-2-4}
}
Changming Ding. Lyapunov quasi-stable trajectories. Fundamenta Mathematicae, Tome 220 (2013) pp. 139-154. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-2-4/