An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces

Abstract

We establish some results that concern the Cauchy-Peano problem in Banach spaces. We first prove that a Banach space contains a nontrivial separable quotient iff its dual admits a weak?-transfinite Schauder frame. We then use this to recover some previous results on quotient spaces. In particular, by applying a recent result of Hájek-Johanis, we find a new perspective for proving the failure of the weak form of Peano's theorem in general Banach spaces. Next, we study a kind of algebraic genericity for the weak form of Peano's theorem in Banach spaces E having complemented subspaces with unconditional Schauder basis. Let K(E) denote the family of all continuous vector fields f : E → E for which u = f(u) has no solutions at any time. It is proved that K(E)u{0} is spaceable in the sense that it contains a closed infinite-dimensional subspace of C(E), the locally convex space of all continuous vector fields on E with the linear topology of uniform convergence on bounded sets. This yields a generalization of a recent result proved for the space c0. We also introduce and study a natural notion of weak-approximate solutions for the nonautonomous Cauchy-Peano problem in Banach spaces. It is proved that the absence of ℓ1-isomorphs inside the underlying space is equivalent to the existence of such approximate solutions. © Instytut Matematyczny PAN, 2013.