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) ∪ {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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm216-3-2,
author = {C. S. Barroso and M. A. M. Marrocos and M. P. Rebou\c cas},
title = {An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces},
journal = {Studia Mathematica},
volume = {215},
year = {2013},
pages = {219-235},
zbl = {1282.34063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm216-3-2}
}
C. S. Barroso; M. A. M. Marrocos; M. P. Rebouças. An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces. Studia Mathematica, Tome 215 (2013) pp. 219-235. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm216-3-2/