On invariant sets and on a theorem of Ważewski

Abstract:

The first part of the paper treats the question of the existence of a solution $x = x(t)$ of an ordinary differential equation which exists for $t \geqq {t_0}$ and remains in a given closed set F for every assigned initial point $({t_0},x({t_0})) \in F$ or, in the autonomous case, $x({t_0}) \in F$. The results involve conditions which, for the autonomous case, reduce to ${\text {dist}}({x^0} + hf({x^0}),F)/h \to 0$ as $h \to + 0$ for all ${x^0} \in F$. The second part of the paper deals with theorems of the Ważewski type which, in some situations, permit the relaxation of the hypothesis that egress points are strict egress points.