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

DOI:
https://doi.org/10.1090/S0002-9939-2012-11154-2

Published electronically:
January 5, 2012

MathSciNet review:
2910769

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[AB]**J. Andres and A. M. Bersani,*Almost-periodicity problem as a fixed-point problem for evolution inclusions.*Topol. Meth. Nonlin. Anal.**18**, 2 (2001), 337-349. MR**1911386 (2003d:34125)****[ABG]**J. Andres, A. M. Bersani and R. F. Grande,*Hierarchy of almost-periodic function spaces.*Rend. Matem. Appl.**26**, 7 (2006), 121-188. MR**2275292 (2008b:43010)****[AP]**J. Andres and D. Pennequin,*On Stepanov almost-periodic oscillations and their discretizations.*J. Differ. Equations Appl., to appear.**[Be]**A. S. Besicovitch,*Almost Periodic Functions.*Dover, New York, 1954. MR**0068029 (16:817a)****[Br]**H. Brézis,*Analyse fonctionnelle, théorie et applications.*Masson, Paris, 1987. MR**697382 (85a:46001)****[C1]**C. Corduneanu,*Almost Periodic Functions.*Interscience Publishers, New York, 1968; Wiley, Chelsea, New York, 1989. MR**0481915 (58:2006)****[C2]**C. Corduneanu,*Some almost periodicity criteria for ordinary differential equations.*Libertas Mathematica**3**(1983), 21-43. MR**722073 (85b:34056)****[C3]**C. Corduneanu,*Almost periodic solutions to differential equations in abstract spaces.*Rev. Roum. Math. Pures Appl.**42**, 9-10 (1997), 753-758. MR**1656645 (99i:34083)****[D1]**L. I. Danilov,*Measure-valued almost periodic functions and almost periodic selections of multivalued maps.*Sb. Math.**188**, 10 (1997), 1417-1438; translation from the Russian original in Mat. Sb.**188**, 10 (1997), 3-24. MR**1485446 (99e:42016)****[Ha]**A. Haraux,*Asymptotic behavior of two-dimensional, quasi-autonomous, almost-periodic evolution equations.*J. Diff. Eqns.**66**, 5 (1987), 62-70. MR**871571 (88a:34061)****[Hu1]**Z. Hu,*Contributions to the Theory of Almost Periodic Differential Equations.*PhD Thesis, Carleton Univ., Ottawa (Ontario, Canada), 2001.**[Hu2]**Z. Hu,*Boundedness and Stepanov almost periodicty of solutions.*Electronic J. Diff. Equations**35**(2005), 1-7. MR**2135246 (2006a:34137)****[HM1]**Z. Hu and A. B. Mingarelli,*On a theorem of Favard.*Proc. Amer. Math. Soc.**132**(2004), 417-428. MR**2022364 (2004i:34120)****[HM2]**Z. Hu and A. B. Mingarelli,*Favard's theorem for almost periodic processes on Banach space.*Proc. Dynam. Syst. Appl.**14**(2005), 615-631. MR**2179169 (2006j:34115)****[HM3]**Z. Hu and A. B. Mingarelli,*Bochner's theorem and Stepanov almost periodic functions.*Ann. Mat. Pura Appl.**187**(2008), 719-736. MR**2413376 (2009d:34115)****[Le]**B. M. Levitan,*Almost-Periodic Functions.*Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953 (in Russian). MR**0060629 (15:700a)****[LZ]**B. M. Levitan and V. V. Zhikov,*Almost-Periodic Functions and Differential Equations.*Cambridge Univ. Press, Cambridge, 1982. MR**690064 (84g:34004)****[Pa]**A. A. Pankov,*Bounded solutions, almost periodic in time, of a class of nonlinear evolution equations.*Mathematics of the USSR - Sbornik**49**, 1 (1984), 73-86; translation from the Russian original in Mat. Sb. (N.S.)**121(163)**, 1 (5) (1983), 72-86. MR**699739 (84i:34081)****[PK]**N.S. Papageorgiou and S. Th. Kyritsi-Yiallourou,*Handbook of Applied Analysis.*Springer, Berlin, 2009. MR**2527754 (2010g:49001)****[Rd]**L. Radová,*Theorems of Bohr-Neugebauer-type for almost-periodic differential equations.*Math. Slovaca**54**, 2 (2004), 169-189. MR**2074215 (2005d:34094)****[Ro]**A. S. Rao,*On differential operators with Bohr-Neugebauer property.*J. Diff. Eqns.**13**(1973), 490-494. MR**0328256 (48:6598)****[Ta]**M. Tarallo,*A Stepanov version for Favard theory.*Arch. Math. (Basel)**90**(2008), 53-59. MR**2382470 (2009a:34016)**

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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:
https://doi.org/10.1090/S0002-9939-2012-11154-2

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.