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Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process
Admissibility and polynomial dichotomies for evolution families
Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia |
For an arbitrary evolution family, we consider the notion of a polynomial dichotomy with respect to a family of norms and characterize it in terms of the admissibility property, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms, we recover the notion of a (strong) nonuniform polynomial dichotomy. As a nontrivial application of the characterization, we establish the robustness of the notion of a strong nonuniform polynomial dichotomy under sufficiently small linear perturbations.
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L. Barreira, D. Dragičević and C. Valls,
Strong and weak $(L^p, L^q)$-admissibility, Bull. Sci. Math., 138 (2014), 721-741.
doi: 10.1016/j.bulsci.2013.11.005. |
| [2] |
L. Barreira, D. Dragičević and C. Valls,
Admissibility on the half line for evolution families, J. Anal. Math., 132 (2017), 157-176.
doi: 10.1007/s11854-017-0017-4. |
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L. Barreira, D. Dragičević and C. Valls, Admissibility and Hyperbolicity, Springer Briefs in Mathematics, Springer, Cham, 2018.
doi: 10.1007/978-3-319-90110-7. |
| [4] |
L. Barreira and C. Valls,
Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst., 22 (2008), 509-528.
doi: 10.3934/dcds.2008.22.509. |
| [5] |
L. Barreira and C. Valls,
Polynomial growth rates, Nonlinear Anal., 71 (2009), 5208-5219.
doi: 10.1016/j.na.2009.04.005. |
| [6] |
L. Barreira and C. Valls,
Robustness of noninvertible dichotomies, J. Math. Soc. Japan, 67 (2015), 293-317.
doi: 10.2969/jmsj/06710293. |
| [7] |
A. Bento and C. Silva,
Stable manifolds for nonuniform polynomial dichotomies, J. Funct. Anal., 257 (2009), 122-148.
doi: 10.1016/j.jfa.2009.01.032. |
| [8] |
A. Bento and C. Silva,
Stable manifolds for nonautonomous equations with nonuniform polynomial dichotomies, Q. J. Math., 63 (2012), 275-308.
doi: 10.1093/qmath/haq047. |
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W. Coppel, Dichotomies in Stability Theory, Lect. Notes. in Math., 629, Springer-Verlag, Berlin-New York, 1979.
doi: 10.1007/BFb0067780. |
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Ju. Dalec'kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, 43, American Mathematical Society, Providence, R.I., 1974. |
| [11] |
D. Dragičević, Admissibility and nonuniform polynomial dichotomies, Math. Nachr., to appear. |
| [12] |
P. V. Hai,
On the polynomial stability of evolution families, Appl. Anal., 95 (2016), 1239-1255.
doi: 10.1080/00036811.2015.1058364. |
| [13] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981
doi: 10.1007/BFb0089647. |
| [14] |
N. T. Huy,
Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354.
doi: 10.1016/j.jfa.2005.11.002. |
| [15] |
Y. Latushkin, T. Randolph and R. Schnaubelt,
Exponential dichotomy and mild solution of nonautonomous equations in Banach spaces, J. Dynam. Differential
Equations, 10 (1998), 489-510.
doi: 10.1023/A:1022609414870. |
| [16] |
T. Li,
Die Stabilitätsfrage bei Differenzengleichungen, Acta Math., 63 (1934), 99-141.
doi: 10.1007/BF02547352. |
| [17] |
N. Lupa and L. H. Popescu,
Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line, Semigroup Forum, 98 (2019), 184-208.
doi: 10.1007/s00233-018-9985-7. |
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J. L. Massera and J. J. Schäffer, Linear differential equations and functional analysis, I, Ann.
of Math. (2), 67 (1958), 517–573.
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J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure
and Applied Mathematics, 21, Academic Press, New York-London, 1966. |
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M. Megan, A. L. Sasu and B. Sasu,
On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory, 44 (2002), 71-78.
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J. S. Muldowney,
Dichotomies and asymptotic behaviour for linear differential systems, Trans. Amer. Math. Soc., 283 (1984), 465-484.
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| [22] |
R. Naulin and M. Pinto,
Roughness of $(h, k)$-dichotomies, J. Differential Equations, 118 (1995), 20-35.
doi: 10.1006/jdeq.1995.1065. |
| [23] |
R. Naulin and M. Pinto,
Stability of discrete dichotomies for linear difference systems, J. Difference Equ. Appl., 3 (1997), 101-123.
doi: 10.1080/10236199708808090. |
| [24] |
O. Perron,
Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
| [25] |
P. Preda and M. Megan,
Nonuniform dichotomy of evolutionary processes in Banach spaces, Bull. Austral. Math. Soc., 27 (1983), 31-52.
doi: 10.1017/S0004972700011473. |
| [26] |
P. Preda, A. Pogan and C. Preda,
$(L^p, L^q)$-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equations Operator Theory, 49 (2004), 405-418.
doi: 10.1007/s00020-002-1268-7. |
| [27] |
P. Preda, A. Pogan and C. Preda,
Schäffer spaces and exponential dichotomy for evolutionary processes, J. Differential Equations, 230 (2006), 378-391.
doi: 10.1016/j.jde.2006.02.004. |
| [28] |
A. L. Sasu, M. G. Babutia and B. Sasu,
Admissibility and nonuniform exponential dichotomy on the half-line, Bull. Sci. Math., 137 (2013), 466-484.
doi: 10.1016/j.bulsci.2012.11.002. |
| [29] |
A. L. Sasu and B. Sasu,
Exponential dichotomy on the real line and admissibility of function spaces, Integral Equations Operator Theory, 54 (2006), 113-130.
doi: 10.1007/s00020-004-1347-z. |
| [30] |
A. L. Sasu and B. Sasu,
Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z., 253 (2006), 515-536.
doi: 10.1007/s00209-005-0920-8. |
| [31] |
A. L. Sasu and B. Sasu,
Integral equations, dichotomy of evolution families on the half-line and applications, Integral Equations Operator Theory, 66 (2010), 113-140.
doi: 10.1007/s00020-009-1735-5. |
| [32] |
N. Van Minh, F. Räbiger and R. Schnaubelt,
Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory, 32 (1998), 332-353.
doi: 10.1007/BF01203774. |
| [33] |
L. Zhou and W. Zhang,
Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Funct. Anal., 271 (2016), 1087-1129.
doi: 10.1016/j.jfa.2016.06.005. |
| [34] |
L. Zhou, K. Lu and W. Zhang,
Equivalences between nonuniform exponential dichotomy and admissibility, J. Differential Equations, 262 (2017), 682-747.
doi: 10.1016/j.jde.2016.09.035. |
show all references
References:
| [1] |
L. Barreira, D. Dragičević and C. Valls,
Strong and weak $(L^p, L^q)$-admissibility, Bull. Sci. Math., 138 (2014), 721-741.
doi: 10.1016/j.bulsci.2013.11.005. |
| [2] |
L. Barreira, D. Dragičević and C. Valls,
Admissibility on the half line for evolution families, J. Anal. Math., 132 (2017), 157-176.
doi: 10.1007/s11854-017-0017-4. |
| [3] |
L. Barreira, D. Dragičević and C. Valls, Admissibility and Hyperbolicity, Springer Briefs in Mathematics, Springer, Cham, 2018.
doi: 10.1007/978-3-319-90110-7. |
| [4] |
L. Barreira and C. Valls,
Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst., 22 (2008), 509-528.
doi: 10.3934/dcds.2008.22.509. |
| [5] |
L. Barreira and C. Valls,
Polynomial growth rates, Nonlinear Anal., 71 (2009), 5208-5219.
doi: 10.1016/j.na.2009.04.005. |
| [6] |
L. Barreira and C. Valls,
Robustness of noninvertible dichotomies, J. Math. Soc. Japan, 67 (2015), 293-317.
doi: 10.2969/jmsj/06710293. |
| [7] |
A. Bento and C. Silva,
Stable manifolds for nonuniform polynomial dichotomies, J. Funct. Anal., 257 (2009), 122-148.
doi: 10.1016/j.jfa.2009.01.032. |
| [8] |
A. Bento and C. Silva,
Stable manifolds for nonautonomous equations with nonuniform polynomial dichotomies, Q. J. Math., 63 (2012), 275-308.
doi: 10.1093/qmath/haq047. |
| [9] |
W. Coppel, Dichotomies in Stability Theory, Lect. Notes. in Math., 629, Springer-Verlag, Berlin-New York, 1979.
doi: 10.1007/BFb0067780. |
| [10] |
Ju. Dalec'kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, 43, American Mathematical Society, Providence, R.I., 1974. |
| [11] |
D. Dragičević, Admissibility and nonuniform polynomial dichotomies, Math. Nachr., to appear. |
| [12] |
P. V. Hai,
On the polynomial stability of evolution families, Appl. Anal., 95 (2016), 1239-1255.
doi: 10.1080/00036811.2015.1058364. |
| [13] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981
doi: 10.1007/BFb0089647. |
| [14] |
N. T. Huy,
Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354.
doi: 10.1016/j.jfa.2005.11.002. |
| [15] |
Y. Latushkin, T. Randolph and R. Schnaubelt,
Exponential dichotomy and mild solution of nonautonomous equations in Banach spaces, J. Dynam. Differential
Equations, 10 (1998), 489-510.
doi: 10.1023/A:1022609414870. |
| [16] |
T. Li,
Die Stabilitätsfrage bei Differenzengleichungen, Acta Math., 63 (1934), 99-141.
doi: 10.1007/BF02547352. |
| [17] |
N. Lupa and L. H. Popescu,
Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line, Semigroup Forum, 98 (2019), 184-208.
doi: 10.1007/s00233-018-9985-7. |
| [18] |
J. L. Massera and J. J. Schäffer, Linear differential equations and functional analysis, I, Ann.
of Math. (2), 67 (1958), 517–573.
doi: 10.2307/1969871. |
| [19] |
J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure
and Applied Mathematics, 21, Academic Press, New York-London, 1966. |
| [20] |
M. Megan, A. L. Sasu and B. Sasu,
On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory, 44 (2002), 71-78.
doi: 10.1007/BF01197861. |
| [21] |
J. S. Muldowney,
Dichotomies and asymptotic behaviour for linear differential systems, Trans. Amer. Math. Soc., 283 (1984), 465-484.
doi: 10.2307/1999142. |
| [22] |
R. Naulin and M. Pinto,
Roughness of $(h, k)$-dichotomies, J. Differential Equations, 118 (1995), 20-35.
doi: 10.1006/jdeq.1995.1065. |
| [23] |
R. Naulin and M. Pinto,
Stability of discrete dichotomies for linear difference systems, J. Difference Equ. Appl., 3 (1997), 101-123.
doi: 10.1080/10236199708808090. |
| [24] |
O. Perron,
Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
| [25] |
P. Preda and M. Megan,
Nonuniform dichotomy of evolutionary processes in Banach spaces, Bull. Austral. Math. Soc., 27 (1983), 31-52.
doi: 10.1017/S0004972700011473. |
| [26] |
P. Preda, A. Pogan and C. Preda,
$(L^p, L^q)$-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equations Operator Theory, 49 (2004), 405-418.
doi: 10.1007/s00020-002-1268-7. |
| [27] |
P. Preda, A. Pogan and C. Preda,
Schäffer spaces and exponential dichotomy for evolutionary processes, J. Differential Equations, 230 (2006), 378-391.
doi: 10.1016/j.jde.2006.02.004. |
| [28] |
A. L. Sasu, M. G. Babutia and B. Sasu,
Admissibility and nonuniform exponential dichotomy on the half-line, Bull. Sci. Math., 137 (2013), 466-484.
doi: 10.1016/j.bulsci.2012.11.002. |
| [29] |
A. L. Sasu and B. Sasu,
Exponential dichotomy on the real line and admissibility of function spaces, Integral Equations Operator Theory, 54 (2006), 113-130.
doi: 10.1007/s00020-004-1347-z. |
| [30] |
A. L. Sasu and B. Sasu,
Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z., 253 (2006), 515-536.
doi: 10.1007/s00209-005-0920-8. |
| [31] |
A. L. Sasu and B. Sasu,
Integral equations, dichotomy of evolution families on the half-line and applications, Integral Equations Operator Theory, 66 (2010), 113-140.
doi: 10.1007/s00020-009-1735-5. |
| [32] |
N. Van Minh, F. Räbiger and R. Schnaubelt,
Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory, 32 (1998), 332-353.
doi: 10.1007/BF01203774. |
| [33] |
L. Zhou and W. Zhang,
Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Funct. Anal., 271 (2016), 1087-1129.
doi: 10.1016/j.jfa.2016.06.005. |
| [34] |
L. Zhou, K. Lu and W. Zhang,
Equivalences between nonuniform exponential dichotomy and admissibility, J. Differential Equations, 262 (2017), 682-747.
doi: 10.1016/j.jde.2016.09.035. |
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