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May  2013, 33(5): 1857-1882. doi: 10.3934/dcds.2013.33.1857

Almost periodic and almost automorphic solutions of linear differential equations

Tomás Caraballo 1, and  David Cheban 2,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla
2. 

State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chişinău

Received  December 2011 Revised  May 2012 Published  December 2012

We analyze the existence of almost periodic (respectively, almost automorphic, recurrent) solutions of a linear non-homogeneous differential (or difference) equation in a Banach space, with almost periodic (respectively, almost automorphic, recurrent) coefficients. Under some conditions we prove that one of the following alternatives is fulfilled:
  (i) There exists a complete trajectory of the corresponding homogeneous equation with constant positive norm;
  (ii) The trivial solution of the homogeneous equation is uniformly asymptotically stable.
If the second alternative holds, then the non-homogeneous equation with almost periodic (respectively, almost automorphic, recurrent) coefficients possesses a unique almost periodic (respectively, almost automorphic, recurrent) solution. We investigate this problem within the framework of general linear nonautonomous dynamical systems. We apply our general results also to the cases of functional-differential equations and difference equations.
Keywords: Almost periodic solution, almost automorphic solutions, cocycle, uniform asymptotic stability, linear nonautonomous contractive dynamical systems., nonautonomous dynamical systems.
Mathematics Subject Classification: 34G10, 34K06, 34K14, 37B20, 37B25, 37B55, 37C75, 37L15, 37L30, 39A05, 39A24, 93D2.
Citation: Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857
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show all references

References:
[1]

Vestnik Moskov. Univ. Ser. I Mat. Meh., 26 (1971), 11-15.  Google Scholar

[2]

Bull. Sci. Math. (2), 101 (1977), 131-148.  Google Scholar

[3]

Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  Google Scholar

[4]

Hermann, Paris, 1955.  Google Scholar

[5]

Noordhoff, 1979.  Google Scholar

[6]

Kishinev, "Shtiintsa", 1984 (in Russian).  Google Scholar

[7]

Comm. Pure Applied Analysis, 11 (2012), 809-828. doi: 10.3934/cpaa.2012.11.809.  Google Scholar

[8]

Bulletin of Academy of Sciences of Republic of Moldova, Mathematics, 2 (1994), 2-21.  Google Scholar

[9]

Electronic Journal of Differential Equations, 2000 (2000), 1-18.  Google Scholar

[10]

Interdisciplinary Mathematical Sciences 1. River Edge, NJ: World Scientific, 2004, 528 pp. doi: 10.1142/9789812563088.  Google Scholar

[11]

Journal of Dynamics and Differential Equations, 20 (2008), 669-697. doi: 10.1007/s10884-008-9101-x.  Google Scholar

[12]

Portugaliae Mathematica, 59 (2002), Fasc. 2, Nova Série, 141-158.  Google Scholar

[13]

Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 203-206.  Google Scholar

[14]

Ann. of Diff. Eqs., 6 (1990), 271-279.  Google Scholar

[15]

Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[16]

Cambridge Univ. Press, London, 1982.  Google Scholar

[17]

Rocky Mountain J. Math., 7 (1977), 231-250.  Google Scholar

[18]

Cambridge University Press. Cambridge - New York - Port Chester - Melbourn - Sydney, 1989, 342 pp.  Google Scholar

[19]

Journal of Differential Equations, 15 (1974), 429-458.  Google Scholar

[20]

Journal of Differential Equations, 113 (1994), 17-67 doi: 10.1006/jdeq.1994.1113.  Google Scholar

[21]

Van Nostrand-Reinbold, London, 1971.  Google Scholar

[22]

volume I. Hermann, 1967. Google Scholar

[23]

Ştiinţa, Chişinău, 1972. (In Russian)  Google Scholar

[24]

Differential Equations, 11 (1975), 1246-1255.  Google Scholar

[25]

An. Şti. Univ. "Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.), 21 (1975), 57-59. (in Russian).  Google Scholar

[26]

Ştiinţa, Chişinău, 1985. (In Russian)  Google Scholar

[27]

Mem. Amer. Math. Soc., 136 (1998), x+93 pp.  Google Scholar

[28]

Noordhoff, Leyden, 1975.  Google Scholar

[29]

Nauka i Tehnika. Minsk, 1986 (in Russian).  Google Scholar

[30]

Lecture Notes in Mathematics, 458, Springer-Verlag, Berlin, 1975, 198 pp.  Google Scholar

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