An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces
Volume 216 / 2013
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$^\star$-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 $\mathscr{K}(E)$ denote the family of all continuous vector fields $f\colon E\to E$ for which $u'=f(u)$ has no solutions at any time. It is proved that $\mathscr{K}(E)\cup \{0\}$ 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 $c_0$. 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 $\ell_1$-isomorphs inside the underlying space is equivalent to the existence of such approximate solutions.
