Generic existence of a solution for a differential equation in a scale of Banach spaces

Abstract:

Let $\left \{ {{X_s}:\alpha \leqslant s \leqslant \beta } \right \}$ be a scale of Banach spaces, $J$ a real interval, $U$ an open subset of $J \times {X_s}$ for some $s$. In this paper we prove that the existence of solutions for \[ x’ = A(t)x + f(t,x),\quad x({t_0}) = {x_0},\] is a generic property, when $A(t)$ is an operator satisfying \[ {\left | {A(t)} \right |_{L({X_{s’}};{X_s})}} \leqslant M{(s’ - s)^{ - 1}}\quad (M > 0\;{\text {independent}}\;{\text {of}}\;s,s’,t)\] in the scale $\left \{ {{X_s}} \right \}$ and $f:J \times U \to {X_\beta }$ is continuous.