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

C. S. Barroso, M. A. M. Marrocos, and M. P. Rebouças. "An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces." Studia Mathematica 216.3 (2013): 219-235. <http://eudml.org/doc/285643>.

@article{C2013,
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) ∪ \{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.},
author = {C. S. Barroso, M. A. M. Marrocos, M. P. Rebouças},
journal = {Studia Mathematica},
keywords = {Peano's theorem; separable quotient problem; -isomorphs; weak compactness; abstract fixed point approximation},
language = {eng},
number = {3},
pages = {219-235},
title = {An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces},
url = {http://eudml.org/doc/285643},
volume = {216},
year = {2013},
}

TY - JOUR
AU - C. S. Barroso
AU - M. A. M. Marrocos
AU - M. P. Rebouças
TI - An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces
JO - Studia Mathematica
PY - 2013
VL - 216
IS - 3
SP - 219
EP - 235
AB - 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.
LA - eng
KW - Peano's theorem; separable quotient problem; -isomorphs; weak compactness; abstract fixed point approximation
UR - http://eudml.org/doc/285643
ER -