A generalization of Peano’s existence theorem and flow invariance

Abstract:

Let $F \subseteq {R^n}$ be closed and $A:F \to {R^n}$ be continuous. Assuming that for $y \in F$ the distance from $y + hAy$ to F is $o(h)$ as $h \downarrow 0$, it is shown that for each $x \in F$ the Cauchy problem $u’ = Au$, $u(0) = x$, has a solution $u:[0,{T_x}] \to F$ on some interval $[0,{T_x}],{T_x} > 0$.