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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Lineability and spaceability for the weak form of Peano's theorem and vector-valued sequence spaces


Authors: Cleon S. Barroso, Geraldo Botelho, Vinícius V. Fávaro and Daniel Pellegrino
Journal: Proc. Amer. Math. Soc. 141 (2013), 1913-1923
MSC (2010): Primary 15A03, 46B45, 34A12
Published electronically: December 28, 2012
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Abstract: Two new applications of a technique for spaceability are given in this paper. For the first time this technique is used in the investigation of the algebraic genericity property of the weak form of Peano's theorem on the existence of solutions of the ODE $ u'=f(u)$ on $ c_0$. The space of all continuous vector fields $ f$ on $ c_0$ is proved to contain a closed $ \mathfrak{c}$-dimensional subspace formed by fields $ f$ for which, except for the null field, the weak form of Peano's theorem fails to be true. The second application generalizes known results on the existence of closed $ \mathfrak{c}$-dimensional subspaces inside certain subsets of $ \ell _p(X)$-spaces, $ 0 < p < \infty $, to the existence of closed subspaces of maximal dimension inside such subsets.


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  • 1. R. M. Aron, F. J. García-Pacheco, D. Pérez-García and J. B. Seoane-Sepúlveda, On dense-lineability of sets of functions on $ \mathbb{R}$, Topology 48 (2009), 149-156. MR 2596209 (2011c:26011)
  • 2. R. M. Aron, V. I. Gurariy and J. B. Seoane-Sepúlveda, Lineability and spaceability of sets of functions on $ \mathbb{R}$, Proc. Amer. Math. Soc. 133 (2005), 795-803. MR 2113929 (2006i:26004)
  • 3. K. Astala, On Peano's theorem in locally convex spaces, Studia Math. 73 (1982), 213-223. MR 675425 (84m:34090)
  • 4. G. Botelho, D. Diniz, V. Fávaro and D. Pellegrino, Spaceability in Banach and quasi-Banach sequence spaces, Linear Algebra Appl. 434 (2011), 1255-1260. MR 2763584
  • 5. A. Cellina, On the nonexistence of solutions of differential equations in nonreflexive spaces, Bull. Amer. Math. Soc. 78 (1972), 1069-1072. MR 0312017 (47:579)
  • 6. J. Dieudonné, Deux exemples singuliers d'équations différentielles, Acta Sci. Math. (Szeged) 12B (1950), 38-40. MR 0035397 (11:729d)
  • 7. J. L. Gámez-Merino, G. A. Muñoz-Fernández, V. M. Sánchez and J. B. Seoane-Sepúlveda, Sierpiński-Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3863-3876. MR 2679609 (2011g:46042)
  • 8. F. J. García-Pacheco and J. B. Seoane-Sepúlveda, Vector spaces of non-measurable functions, Acta Math. Sin. (Engl. Ser.) 22 (2006), 1805-1808. MR 2262440 (2007i:28006)
  • 9. A. N. Godunov, Counterexample to Peano's theorem in infinite-dimensional Hilbert space, Vestnik Mosko. Univ. Ser. 1 Mat. Mekh. [Moscow Univ. Math. Bull.] 5 (1972), 19-21.
  • 10. A. N. Godunov, On Peano's theorem in Banach spaces, Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.] 9 (1975), 59-60. MR 0364797 (51:1051)
  • 11. P. Hájek and M. Johanis, On Peano's theorem in Banach spaces, J. Differential Equations 249 (2010), 3342-3351. MR 2737433 (2011j:34182)
  • 12. D. Kitson and R. M. Timoney, Operator ranges and spaceability, J. Math. Anal. Appl. 378 (2011), 680-686. MR 2773276
  • 13. A. Lasota and J. A. Yorke, The generic property of existence of solutions of differential equations in Banach spaces, J. Differential Equations 13 (1973), 1-12. MR 0335994 (49:770)
  • 14. S. G. Lobanov, Peano's theorem is invalid for any infinite-dimensional Fréchet space (Russian), Mat. Sb. 184 (1993), 83-86; translation in Russian Acad. Sci. Sb. Math. 78 (1994), 211-214. MR 1214945 (93m:34095)
  • 15. S. G. Lobanov and O. G. Smolyanov, Ordinary differential equations in locally convex spaces (Russian), Uspekhi Mat. Nauk. 49 (1994), 93-168; translation in Russian Math. Surveys 49 (1994), 97-175. MR 1289388 (95k:34002)
  • 16. J. López-Salazar, Vector spaces of entire functions of unbounded type, Proc. Amer. Math. Soc. 139 (2011), 1347-1360. MR 2748427
  • 17. G. Metafune and V. B. Moscatelli, On the space $ l^{p+}=\bigcap _{q>p}l^{q}$, Math. Nachr. 147 (1990), 7-12. MR 1127304 (92i:46005)
  • 18. S. A. Shkarin, Peano's theorem in infinite-dimensional Fréchet spaces is invalid (Russian), Funktsional. Anal. i Prilozhen. 27 (1993), 90-92; translation in Funct. Anal. Appl. 27 (1993), 149-151. MR 1251174 (94h:34080)
  • 19. S. A. Shkarin, Peano's theorem is invalid in infinite-dimensional $ F'$-spaces (Russian), Mat. Zametki 62 (1997), 128-137; translation in Math. Notes 62 (1997), 108-115. MR 1620008 (99c:34140)
  • 20. J. A. Yorke, A continuous differential equation in Hilbert space without existence, Funkcial. Ekvac. 13 (1970), 19-21. MR 0264196 (41:8792)

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Additional Information

Cleon S. Barroso
Affiliation: Departamento de Matemática, Campus do Pici, Universidade Federal do Ceará, 60.455-760 Fortaleza, Brazil
Email: cleonbar@mat.ufc.br

Geraldo Botelho
Affiliation: Faculdade de Matemática, Universidade Federal de Uberlândia, 38.400-902, Uberlândia, Brazil
Email: botelho@ufu.br

Vinícius V. Fávaro
Affiliation: Faculdade de Matemática, Universidade Federal de Uberlândia, 38.400-902, Uberlândia, Brazil
Email: vvfavaro@gmail.com

Daniel Pellegrino
Affiliation: Departamento de Matemática, Universidade Federal da Paraíba, 58.051-900, João Pessoa, Brazil
Email: pellegrino@pq.cnpq.br, dmpellegrino@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11466-2
PII: S 0002-9939(2012)11466-2
Received by editor(s): June 2, 2011
Received by editor(s) in revised form: September 24, 2011
Published electronically: December 28, 2012
Additional Notes: The first author was supported by CNPq Grant 307210/2009-0.
The second author was supported by CNPq Grant 306981/2008-4.
The third author was supported by FAPEMIG Grant CEX-APQ-00208-09.
The fourth author was supported by CNPq Grant 301237/2009-3 and CAPES-NF
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2012 American Mathematical Society