Eduardo V. Teixeira 
Strong solutions for differential equations in abstract spaces
(402K, AMS-TeX)

ABSTRACT.  Let $(E, \mathcal{F})$ be a locally convex space. We denote the bounded elements of $E$ by $E_b := \{ x \in E :
\sup\limits_{\rho \in \mathcal{F}} \rho(x) < \infty \}$. In this paper we prove that if $ B_{E_b}$ is relatively
compact with respect to the $\mathcal{F}$ topology and $f : I \times E_b \to E_b$ is a measurable family of
$\mathcal{F}$-continuous maps then for each $x_0 \in E_b$ there exists a norm-differentiable local solution to
the Initial Valued Problem $u_t(t) = f(t,u(t))$, $u(t_0) = x_0$. Our final goal is to study the Lipschitz
stability of a differential equation involving the Hardy-Littlewood maximal operator.