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2013 | 216 | 3 | 219-235
Tytuł artykułu

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

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EN
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.
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  • Departamento de Matemática, Universidade Federal do Ceará, Avenida Humberto Monte S/N, Bl 914, 60455-760, Fortaleza, Brazil
  • Departamento de Matemática, Universidade Federal do Amazonas, Av. Rodrigo Otávio Jordão Ramos, ICE, 3000, 3077-000, Manaus, Brazil
  • Departamento de Matemática, Universidade Federal do Amazonas, Av. Rodrigo Otávio Jordão Ramos, ICE, 3000, 3077-000, Manaus, Brazil
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm216-3-2
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