On generic asymptotic stability of differential equations in Banach space

Abstract:

The asymptotic stability of the zero solution of the differential equation $( \ast )\;x’ = Ax + f(x)$ is studied, when the pertubation f is in a given complete metric space $\mathfrak {M}$. It is known that the zero solution of $( \ast )$ is asymptotically stable whenever f is in a certain proper subset $\mathfrak {N} \subset \mathfrak {M}$. It is shown that, while $\mathfrak {N}$ is of Baire first category in $\mathfrak {M}$, on the contrary the set ${\mathfrak {M}_0}$ of all those f for which the zero solution of $( \ast )$ is asymptotically stable is a proper residual subset of $\mathfrak {M}$.