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).

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.

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