We define and characterize in Banach spaces the property of oscillation of a semidynamical system at the neighbourhood of a fixed point. The main idea of our investigation is to show that there does not exist a normal cone which would contain a nontrivial trajectory.
1998 Academic Press
Let E be a Banach space and let f : E Γ E. We assume that a # E is a fixed point of f. The famous Hadamard Perron and Grobman Hartman Theorems describe the behaviour of dynamical systems at the neighbourhood of a for which the spectrum of Df (a) lies off the unit circle. Our aim is to investigate the case when Df (a) has no eigenvalues in R + (Df (a) does not have to be hyperbolic). We show that in this case there arises a kind of oscillation. Therefore, we define and characterize the property of oscillation of semidynamical systems (with discrete and continuous time) in the vicinity of the fixed point.
Our results are closely connected with the theory of positive operators (cf. [6], ). In fact, we investigate operators which are not positive for any normal cone V. Now we are going to establish some notation. By R + we understand the left closed real half-line and by N the set of all nonnegative integers. T stands for N or R + .
Definition 1. Let X be a set, let T # T and let g: T_X Γ X be a semidynamical system. For Y/X we define Inv( g, Y