 04138 Eduardo V. Teixeira
 Strong solutions for differential equations in abstract spaces
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Apr 29, 04

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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 normdifferentiable 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 HardyLittlewood maximal operator.
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