Preliminaries

Abstract

Throughout this lecture, real systems of differential equations will be considered and the following notations will be used. The real intervals a < t < b, a ≤ t ≤ b, a ≤ t < b and a < t ≤ b will be denoted by (a,b), [a,b], [a,b) and (a,b], respectively. Let R denote the whole real line, i.e., R = (-∞,∞) and I denote the interval 0 ≤ t < ∞ and Rⁿ denote Euclidean n-space. For x ε Rⁿ, |x| be any norm of x. For an n × n matrix A = (a_ij), define the norm |A| of A by $\left| {\text{A}} \right|{\text{ = }}\mathop {\sup }\limits_{\left| {\text{X}} \right|{\text{ = 1}}} {\text{ = }}\left| {{\text{AX}}} \right|$, where x ε Rⁿ. The closure of a set S will be denoted by $\overline S$, and N(ε,S) represents the ε-neigh-brohood of S. We shall denote by C(J × D, Rⁿ) the set of all continuous functions f defined on J × D with values in Rⁿ, where J is a subset of R and D is a subset of Rⁿ.