Strong solutions for differential equations in abstract spaces

Let (E, F) be a locally convex space. We denote the bounded elements of E by E-b : ={x is an element of E : parallel to x parallel to(F) = sup rho(x) < infinity}. In this paper, we prove that if B-Eb is relatively compact with respect to the F topology and f : I x Eb --+ El, is a measurable family of F-continuous maps then for each x(0) is an element of E-b there exists a norm-differentiable, (i.e. differentiable with respect to the parallel to (.) parallel to(F) norm) local solution to the initial valued problem u(t)(t) = f(t, u(t)), u(t(0)) = x(0). All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy-Littlewood maximal operator. (c) 2004 Elsevier Inc. All rights reserved.