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

 

On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations


Authors: Jan Andres and Denis Pennequin
Journal: Proc. Amer. Math. Soc. 140 (2012), 2825-2834
MSC (2010): Primary 34C27; Secondary 34C15, 34G20
Published electronically: January 5, 2012
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Abstract: It is shown that in uniformly convex Banach spaces, Stepanov almost-periodic functions with Stepanov almost-periodic derivatives are uniformly almost-periodic in the sense of Bohr. This in natural situations yields, jointly with the derived properties of the associated Nemytskii operators, the nonexistence of purely (i.e.nonuniformly continuous) Stepanov almost-periodic solutions of ordinary differential equations. In particular, the existence problem of such solutions, considered in a series of five papers of Z. Hu and A. B. Mingarelli, is answered in a negative way.


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Additional Information

Jan Andres
Affiliation: Department of Mathematical Analysis, Faculty of Science, Palacký University, Tř. 17 listopadu 12, 771 46 Olomouc, Czech Republic
Email: andres@inf.upol.cz

Denis Pennequin
Affiliation: Université Paris I Panthéon-Sorbonne, Centre PMF, Laboratoire SAMM, 90, Rue de Tolbiac, 75 634 Paris Cedex 13, France
Email: pennequi@univ-paris1.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11154-2
PII: S 0002-9939(2012)11154-2
Keywords: Stepanov almost-periodic solutions, Nemytskii operators, ordinary differential equations, nonexistence results.
Received by editor(s): July 1, 2010
Received by editor(s) in revised form: March 21, 2011
Published electronically: January 5, 2012
Additional Notes: The first author was supported by the Council of Czech Government (MSM 6198959214)
The second author was supported by ANR ANAR
Communicated by: Yingfei Yi
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.