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

C. S. Barroso¹ ; M. A. M. Marrocos² ; M. P. Rebouças²

¹ Departamento de Matemática Universidade Federal do Ceará Avenida Humberto Monte S/N, Bl 914 60455-760, Fortaleza, Brazil
² Departamento de Matemática Universidade Federal do Amazonas Av. Rodrigo Otávio Jordão Ramos ICE, 3000, 3077-000, Manaus, Brazil

Studia Mathematica, Tome 216 (2013) no. 3, pp. 219-235

Résumé

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.

Détail

DOI : 10.4064/sm216-3-2

Keywords: establish results concern cauchy peano problem banach spaces first prove banach space contains nontrivial separable quotient its dual admits weak star transfinite schauder frame recover previous results quotient spaces particular applying recent result jek johanis perspective proving failure weak form peanos theorem general banach spaces study kind algebraic genericity weak form peanos theorem banach spaces having complemented subspaces unconditional schauder basis mathscr denote family continuous vector fields colon which has solutions time proved mathscr cup spaceable sense contains closed infinite dimensional subspace locally convex space continuous vector fields linear topology uniform convergence bounded sets yields generalization recent result proved space introduce study natural notion weak approximate solutions nonautonomous cauchy peano problem banach spaces proved absence ell isomorphs inside underlying space equivalent existence approximate solutions