Strong solutions for differential equations in abstract spaces

Let (E,F) be a locally convex space. We denote the bounded elements of E by Eb:={x∈E:∥x∥F=supρ∈Fρ(x)<∞}. In this paper, we prove that if BEb is relatively compact with respect to the F topology and f:I×Eb→Eb is a measurable family of F-continuous maps then for each x0∈Eb there exists a norm-differentiable, (i.e. differentiable with respect to the ∥·∥F norm) local solution to the initial valued problem ut(t)=f(t,u(t)), u(t0)=x0. All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy-Littlewood maximal operator.