Uniform stability and semi-stability of motions in dynamical systems on metric spaces

Abstract

Some stability properties of motions in pseudo-dynamical systems and semi-systems are studied. Introduction. The purpose of the present paper is to study some properties of motions in dynamical and pseudo-dynamical systems (see definitions below) on metric spaces. We consider uniform stability and semistability (see Section 2) of motions in nonempty subsets of the phase space and discuss in particular properties of mappings of the types: x {the limit set of x} and x {the prolongational limit set of x}, as well as some regularity properties of mappings whose values are some generalized prolongational limit sets (see the formula (1.8)). In Sections 3 and 4 we consider the problem of propagation of stability and uniform stability of motions, from sets onto closures of those sets. In Section 7 we consider certain Lyapunov type functions. The paper is closed by some remarks concerning stability properties of motions preserved when we pass from one dynamical system to another, assuming that those systems satisfy certain conditions of the type of “asymptotic equivalence” in the Wazewski sense. 1. Preliminaries. In order to exclude any misunderstanding we recall fundamental definitions and fix notation used in the sequel. 1991 Mathematics Subject Classification: 34C35, 54H20, 58F10, 58F25, 34D05, 34D20.