March  2020, 19(3): 1321-1336. doi: 10.3934/cpaa.2020064

Admissibility and polynomial dichotomies for evolution families

Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia

Received  March 2019 Revised  July 2019 Published  November 2019

Fund Project: The author is supported by the University of Rijeka under the project uniri-prirod-18-9

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.

Citation: Davor Dragičević. Admissibility and polynomial dichotomies for evolution families. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1321-1336. doi: 10.3934/cpaa.2020064
References:
[1]

L. BarreiraD. 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.  Google Scholar

[2]

L. BarreiraD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. Barreira and C. Valls, Polynomial growth rates, Nonlinear Anal., 71 (2009), 5208-5219.  doi: 10.1016/j.na.2009.04.005.  Google Scholar

[6]

L. Barreira and C. Valls, Robustness of noninvertible dichotomies, J. Math. Soc. Japan, 67 (2015), 293-317.  doi: 10.2969/jmsj/06710293.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[11]

D. Dragičević, Admissibility and nonuniform polynomial dichotomies, Math. Nachr., to appear. Google Scholar

[12]

P. V. Hai, On the polynomial stability of evolution families, Appl. Anal., 95 (2016), 1239-1255.  doi: 10.1080/00036811.2015.1058364.  Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981 doi: 10.1007/BFb0089647.  Google Scholar

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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.  Google Scholar

[15]

Y. LatushkinT. 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.  Google Scholar

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T. Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math., 63 (1934), 99-141.  doi: 10.1007/BF02547352.  Google Scholar

[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.  Google Scholar

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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. doi: 10.2307/1969871.  Google Scholar

[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.  Google Scholar

[20]

M. MeganA. 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.  Google Scholar

[21]

J. S. Muldowney, Dichotomies and asymptotic behaviour for linear differential systems, Trans. Amer. Math. Soc., 283 (1984), 465-484.  doi: 10.2307/1999142.  Google Scholar

[22]

R. Naulin and M. Pinto, Roughness of $(h, k)$-dichotomies, J. Differential Equations, 118 (1995), 20-35.  doi: 10.1006/jdeq.1995.1065.  Google Scholar

[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.  Google Scholar

[24]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.  Google Scholar

[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.  Google Scholar

[26]

P. PredaA. 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.  Google Scholar

[27]

P. PredaA. 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.  Google Scholar

[28]

A. L. SasuM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

N. Van MinhF. 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.  Google Scholar

[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.  Google Scholar

[34]

L. ZhouK. 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.  Google Scholar

show all references

References:
[1]

L. BarreiraD. 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.  Google Scholar

[2]

L. BarreiraD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. Barreira and C. Valls, Polynomial growth rates, Nonlinear Anal., 71 (2009), 5208-5219.  doi: 10.1016/j.na.2009.04.005.  Google Scholar

[6]

L. Barreira and C. Valls, Robustness of noninvertible dichotomies, J. Math. Soc. Japan, 67 (2015), 293-317.  doi: 10.2969/jmsj/06710293.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

W. Coppel, Dichotomies in Stability Theory, Lect. Notes. in Math., 629, Springer-Verlag, Berlin-New York, 1979. doi: 10.1007/BFb0067780.  Google Scholar

[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.  Google Scholar

[11]

D. Dragičević, Admissibility and nonuniform polynomial dichotomies, Math. Nachr., to appear. Google Scholar

[12]

P. V. Hai, On the polynomial stability of evolution families, Appl. Anal., 95 (2016), 1239-1255.  doi: 10.1080/00036811.2015.1058364.  Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981 doi: 10.1007/BFb0089647.  Google Scholar

[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.  Google Scholar

[15]

Y. LatushkinT. 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.  Google Scholar

[16]

T. Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math., 63 (1934), 99-141.  doi: 10.1007/BF02547352.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. MeganA. 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.  Google Scholar

[21]

J. S. Muldowney, Dichotomies and asymptotic behaviour for linear differential systems, Trans. Amer. Math. Soc., 283 (1984), 465-484.  doi: 10.2307/1999142.  Google Scholar

[22]

R. Naulin and M. Pinto, Roughness of $(h, k)$-dichotomies, J. Differential Equations, 118 (1995), 20-35.  doi: 10.1006/jdeq.1995.1065.  Google Scholar

[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.  Google Scholar

[24]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.  Google Scholar

[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.  Google Scholar

[26]

P. PredaA. 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.  Google Scholar

[27]

P. PredaA. 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.  Google Scholar

[28]

A. L. SasuM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

N. Van MinhF. 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.  Google Scholar

[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.  Google Scholar

[34]

L. ZhouK. 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.  Google Scholar

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