Strong solutions for differential equations in abstract spaces

Let be a locally convex space. We denote the bounded elements of E by . In this paper, we prove that if BEb is relatively compact with respect to the topology and f:I×Eb→Eb is a measurable family of -continuous maps then for each x0 Eb there exists a norm-differentiable, (i.e. differentiable with respect to the 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.