November  2013, 33(11&12): 4795-4810. doi: 10.3934/dcds.2013.33.4795

Floquet representations and asymptotic behavior of periodic evolution families

1. 

Universität Tübingen, Mathematisch-Naturwissenschaftliche Fakultät, Auf der Morgenstelle 10, D-72076 Tübingen, Germany, Germany, Germany

Received  August 2011 Revised  September 2011 Published  May 2013

We use semigroup techniques to describe the asymptotic behavior of contractive, periodic evolution families on Hilbert spaces. In particular, we show that such evolution families converge almost weakly to a Floquet representation with discrete spectrum.
Citation: Fatih Bayazit, Ulrich Groh, Rainer Nagel. Floquet representations and asymptotic behavior of periodic evolution families. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4795-4810. doi: 10.3934/dcds.2013.33.4795
References:
[1]

H. Amann, "Gewöhnliche Differentialgleichungen,", de Gruyter, (1983).   Google Scholar

[2]

H. Amann, "Ordinary Differential Equations: An Introduction to Nonlinear Analysis,", de Gruyter, (1990).  doi: 10.1515/9783110853698.  Google Scholar

[3]

H. Bercovici, C. Foiaş, L. Kèrchy and B. Nagy, "Harmonic Analysis of Operators on Hilbert Space,", Springer-Verlag, (2010).   Google Scholar

[4]

C. Chicone, "Ordinary Differential Equations with Applications,", Springer-Verlag, (2006).   Google Scholar

[5]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", American Mathematical Society, (1999).   Google Scholar

[6]

J. Conway, "A Course in Functional Analysis,", Springer-Verlag, (1997).   Google Scholar

[7]

D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Applications,", Pitman Research Notes, (1992).   Google Scholar

[8]

N. Dunford and J. Schwartz, "Linear Operators Part I: General Theory,", Interscience, (1958).   Google Scholar

[9]

T. Eisner, Embedding operators into strongly continuous semigroups,, Archiv Math., 92 (2009), 451.  doi: 10.1007/s00013-009-3154-x.  Google Scholar

[10]

T. Eisner, "Stability of Operators and $C_0$-Semigroups,", Birkhäuser-Verlag, (2010).   Google Scholar

[11]

K.-J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000).   Google Scholar

[12]

G. Floquet, "Sur la Théorie des Équations Différentielles Linèaires,", Gauthier-Villars, (1879).   Google Scholar

[13]

G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques,, Ann. Sci. Ècole Norm. Sup., 12 (1883), 47.   Google Scholar

[14]

J. A. Goldstein, "Semigroups of Operators and Applications,", Oxford University Press, (1985).   Google Scholar

[15]

M. Haase, Spectral properties of operator logarithms,, Math. Z., 245 (2003), 761.  doi: 10.1007/s00209-003-0569-0.  Google Scholar

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Springer-Verlag, (1981).   Google Scholar

[17]

H. Heuser, "Funktionalanalysis - Theorie und Anwendungen,", Teubner-Verlag, (2006).   Google Scholar

[18]

H. Junghenn, Tensor products and almost periodicity,, Proc. Amer. Math. Soc., 43 (1974), 99.  doi: 10.1090/S0002-9939-1974-0365223-3.  Google Scholar

[19]

R. Nagel, Semigroup methods for nonautonomous Cauchy problems,, Evolution Equations, 168 (1995), 301.   Google Scholar

[20]

H. H. Schaefer, "Banach Lattices and Positive Operators,", Springer-Verlag, (1974).   Google Scholar

[21]

R. Schnaubelt, Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations,, Prog. Nonlinear Differential Equations Appl., 50 (2002), 311.   Google Scholar

[22]

A. Stokes, A Floquet theory for functional differential equation,, Proc. Nat. Acad. Sciences USA, 48 (1962), 1330.  doi: 10.1073/pnas.48.8.1330.  Google Scholar

show all references

References:
[1]

H. Amann, "Gewöhnliche Differentialgleichungen,", de Gruyter, (1983).   Google Scholar

[2]

H. Amann, "Ordinary Differential Equations: An Introduction to Nonlinear Analysis,", de Gruyter, (1990).  doi: 10.1515/9783110853698.  Google Scholar

[3]

H. Bercovici, C. Foiaş, L. Kèrchy and B. Nagy, "Harmonic Analysis of Operators on Hilbert Space,", Springer-Verlag, (2010).   Google Scholar

[4]

C. Chicone, "Ordinary Differential Equations with Applications,", Springer-Verlag, (2006).   Google Scholar

[5]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", American Mathematical Society, (1999).   Google Scholar

[6]

J. Conway, "A Course in Functional Analysis,", Springer-Verlag, (1997).   Google Scholar

[7]

D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Applications,", Pitman Research Notes, (1992).   Google Scholar

[8]

N. Dunford and J. Schwartz, "Linear Operators Part I: General Theory,", Interscience, (1958).   Google Scholar

[9]

T. Eisner, Embedding operators into strongly continuous semigroups,, Archiv Math., 92 (2009), 451.  doi: 10.1007/s00013-009-3154-x.  Google Scholar

[10]

T. Eisner, "Stability of Operators and $C_0$-Semigroups,", Birkhäuser-Verlag, (2010).   Google Scholar

[11]

K.-J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000).   Google Scholar

[12]

G. Floquet, "Sur la Théorie des Équations Différentielles Linèaires,", Gauthier-Villars, (1879).   Google Scholar

[13]

G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques,, Ann. Sci. Ècole Norm. Sup., 12 (1883), 47.   Google Scholar

[14]

J. A. Goldstein, "Semigroups of Operators and Applications,", Oxford University Press, (1985).   Google Scholar

[15]

M. Haase, Spectral properties of operator logarithms,, Math. Z., 245 (2003), 761.  doi: 10.1007/s00209-003-0569-0.  Google Scholar

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Springer-Verlag, (1981).   Google Scholar

[17]

H. Heuser, "Funktionalanalysis - Theorie und Anwendungen,", Teubner-Verlag, (2006).   Google Scholar

[18]

H. Junghenn, Tensor products and almost periodicity,, Proc. Amer. Math. Soc., 43 (1974), 99.  doi: 10.1090/S0002-9939-1974-0365223-3.  Google Scholar

[19]

R. Nagel, Semigroup methods for nonautonomous Cauchy problems,, Evolution Equations, 168 (1995), 301.   Google Scholar

[20]

H. H. Schaefer, "Banach Lattices and Positive Operators,", Springer-Verlag, (1974).   Google Scholar

[21]

R. Schnaubelt, Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations,, Prog. Nonlinear Differential Equations Appl., 50 (2002), 311.   Google Scholar

[22]

A. Stokes, A Floquet theory for functional differential equation,, Proc. Nat. Acad. Sciences USA, 48 (1962), 1330.  doi: 10.1073/pnas.48.8.1330.  Google Scholar

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