October  2021, 20(10): 3419-3443. doi: 10.3934/cpaa.2021112

Admissibility and generalized nonuniform dichotomies for discrete dynamics

Universidade da Beira Interior, Rua Marquês d'Ávila e Bolama, 6201-001 Covilhã, Portugal

Received  January 2021 Revised  May 2021 Published  October 2021 Early access  June 2021

Fund Project: The author was partially supported by FCT through CMUBI (project UIDB/MAT/00212/2020)

We obtain characterizations of nonuniform dichotomies, defined by general growth rates, based on admissibility conditions. Additionally, we use the obtained characterizations to derive robustness results for the considered dichotomies. As particular cases, we recover several results in the literature concerning nonuniform exponential dichotomies and nonuniform polynomial dichotomies as well as new results for nonuniform dichotomies with logarithmic growth.

Citation: César M. Silva. Admissibility and generalized nonuniform dichotomies for discrete dynamics. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3419-3443. doi: 10.3934/cpaa.2021112
References:
[1]

M. G. Babuţia, M. Megan and I. L. Popa, On $(h, k)$-dichotomies for nonautonomous linear difference equations in Banach spaces, Int. J. Differ. Equ., (2013), 7 pp. doi: 10.1155/2013/761680.  Google Scholar

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L. BarreiraJ. Chu and C. Valls, Lyapunov functions for general nonuniform dichotomies, Milan J. Math., 81 (2013), 153-169.  doi: 10.1007/s00032-013-0198-y.  Google Scholar

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L. BarreiraD. Dragičević and C. Valls, Admissibility and nonuniformly hyperbolic sets, Electron. J. Qual. Theory Differ. Equ., 10 (2016), 1-15.  doi: 10.14232/ejqtde.2016.1.10.  Google Scholar

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L. BarreiraD. Dragičević and C. Valls, Nonuniform hyperbolicity and one-sided admissibility, Rend. Lincei Mat. Appl., 27 (2016), 1-13.  doi: 10.4171/rlm/732.  Google Scholar

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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

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L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-74775-8.  Google Scholar

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

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A. J. G. Bento and C. M. Silva, Generalized nonuniform dichotomies and local stable manifolds, J. Dynam. Differ. Equ., 25 (2013), 1139-1158.  doi: 10.1007/s10884-013-9331-4.  Google Scholar

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A. J. G. 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. J. G. Bento and C. M. Silva, Stable manifolds for non-autonomous equations with non-uniform polynomial dichotomies, Q. J. Math., 63 (2012), 275-308.  doi: 10.1093/qmath/haq047.  Google Scholar

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X. ChangJ. Zhang and J. Qin, Robustness of nonuniform $(\mu, \nu)$-dichotomies in Banach spaces, J. Math. Anal. Appl., 387 (2012), 582-594.  doi: 10.1016/j.jmaa.2011.09.026.  Google Scholar

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C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monogr. 70, Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/surv/070.  Google Scholar

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W. Coppel, Dichotomies in Stability Theory, Springer, New York, 1981.  Google Scholar

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M. O. Czarnecki and L. Rifford, Approximation and regularization of Lipschitz functions: convergence of the gradients, Trans. Amer. Math. Soc., 358 (2006), 4467-4520.  doi: 10.1090/S0002-9947-06-04103-1.  Google Scholar

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Ju. L. Daleckiǐand M. G. Kreǐn, Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monogr. 43, Amer. Math. Soc., Providence, RI, 1974.  Google Scholar

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D. Dragičević, Admissibility and polynomial dichotomies for evolution families, Commun. Pure Appl. Anal., 19 (2020), 1321-1336.  doi: 10.3934/cpaa.2020064.  Google Scholar

[18]

D. Dragičević, Admissibility and nonuniform polynomial dichotomies, Math. Nachr., 293 (2019), 226-243.  doi: 10.1002/mana.201800291.  Google Scholar

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N. Lupa and M. Megan, Exponential dichotomies of evolution operators in Banach spaces, Monatsh. Math., 174 (2014), 265-284.  doi: 10.1007/s00605-013-0517-y.  Google Scholar

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J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Academic, New York, 1966.  Google Scholar

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Ng uyen Van MinhF. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line, Integral Equ. Oper. Theory, 32 (1998), 332-353.  doi: 10.1007/BF01203774.  Google Scholar

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

[23]

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

[24]

Y. Pesin, Families of invariant manifolds that corresponding to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976) 1332–1379, (Russian) English transl. Math. USSR-Izv. 10 (1976), 1261–1305.  Google Scholar

[25]

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russ. Math. Surv., 32 (1977), 55-114.   Google Scholar

[26]

Y. Pesin, Geodesic flows in closed Riemannian manifolds without focal points, Math. USSR-Izv., 11 (1977), 1195-1228.   Google Scholar

[27]

M. Pinto, Discrete dichotomies, Comput. Math. Appl., 28 (1994), 259-270.  doi: 10.1016/0898-1221(94)00114-6.  Google Scholar

[28]

P. PredaA. Pogan and C. Preda, $(L^p; L^q)$-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equ. Oper. Theory, 49 (2004), 405-418.  doi: 10.1007/s00020-002-1268-7.  Google Scholar

[29]

C. PredaP. Preda and A. Craciunescu, A version of a theorem of R. Datko for nonuniform exponential contractions, J. Math. Anal. Appl., 385 (2012), 572-581.  doi: 10.1016/j.jmaa.2011.06.082.  Google Scholar

[30]

C. PredaP. Preda and A. Craciunescu, Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations, J. Funct. Anal., 258 (2010), 729-757.  doi: 10.1016/j.jfa.2009.09.002.  Google Scholar

[31]

B. Sasu and A. L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 33 (2013), 3057-3084.  doi: 10.3934/dcds.2013.33.3057.  Google Scholar

[32]

A. L. SasuM. G. Babuţia 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

[33]

A. L. Sasu and B. Sasu, Admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line, Appl. Math. Comput., 378 (2020), 125167. doi: 10.1016/j. amc. 2020.125167.  Google Scholar

[34]

B. Sasu and A. L. Sasu, A. L. Sasu and B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces, Integral Equ. Oper. Theory, 54 (2006), 113-130.  doi: 10.1007/s00020-004-1347-z.  Google Scholar

[35]

B. Sasu and A. L. Sasu, Integral equations in the study of the asymptotic behavior of skew-product flows, Asymptotic Anal., 68 (2010), 135-153.   Google Scholar

[36]

L. ZhouK. Lu and W. Zhang, Equivalences between nonuniform exponential dichotomy and admissibility, J. Differ. Equ., 262 (2017), 682-747.  doi: 10.1016/j.jde.2016.09.035.  Google Scholar

[37]

L. Zhou and W. Zhang, Exponential dichotomy and $(\ell^p, \ell^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408.  doi: 10.1016/j.jmaa.2005.04.047.  Google Scholar

show all references

References:
[1]

M. G. Babuţia, M. Megan and I. L. Popa, On $(h, k)$-dichotomies for nonautonomous linear difference equations in Banach spaces, Int. J. Differ. Equ., (2013), 7 pp. doi: 10.1155/2013/761680.  Google Scholar

[2]

L. BarreiraJ. Chu and C. Valls, Lyapunov functions for general nonuniform dichotomies, Milan J. Math., 81 (2013), 153-169.  doi: 10.1007/s00032-013-0198-y.  Google Scholar

[3]

L. Barreira, D. Dragičević and C. Valls, Admissibility and Hyperbolicity, Springerbriefs In Mathematics, Springer, 2018. doi: 10.1007/978-3-319-90110-7.  Google Scholar

[4]

L. BarreiraD. Dragičević and C. Valls, Admissibility and nonuniformly hyperbolic sets, Electron. J. Qual. Theory Differ. Equ., 10 (2016), 1-15.  doi: 10.14232/ejqtde.2016.1.10.  Google Scholar

[5]

L. BarreiraD. Dragičević and C. Valls, Nonuniform hyperbolicity and one-sided admissibility, Rend. Lincei Mat. Appl., 27 (2016), 1-13.  doi: 10.4171/rlm/732.  Google Scholar

[6]

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

[7]

L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-74775-8.  Google Scholar

[8]

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

[9]

A. J. G. Bento and C. M. Silva, Generalized nonuniform dichotomies and local stable manifolds, J. Dynam. Differ. Equ., 25 (2013), 1139-1158.  doi: 10.1007/s10884-013-9331-4.  Google Scholar

[10]

A. J. G. 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

[11]

A. J. G. Bento and C. M. Silva, Stable manifolds for non-autonomous equations with non-uniform polynomial dichotomies, Q. J. Math., 63 (2012), 275-308.  doi: 10.1093/qmath/haq047.  Google Scholar

[12]

X. ChangJ. Zhang and J. Qin, Robustness of nonuniform $(\mu, \nu)$-dichotomies in Banach spaces, J. Math. Anal. Appl., 387 (2012), 582-594.  doi: 10.1016/j.jmaa.2011.09.026.  Google Scholar

[13]

C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monogr. 70, Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/surv/070.  Google Scholar

[14]

W. Coppel, Dichotomies in Stability Theory, Springer, New York, 1981.  Google Scholar

[15]

M. O. Czarnecki and L. Rifford, Approximation and regularization of Lipschitz functions: convergence of the gradients, Trans. Amer. Math. Soc., 358 (2006), 4467-4520.  doi: 10.1090/S0002-9947-06-04103-1.  Google Scholar

[16]

Ju. L. Daleckiǐand M. G. Kreǐn, Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monogr. 43, Amer. Math. Soc., Providence, RI, 1974.  Google Scholar

[17]

D. Dragičević, Admissibility and polynomial dichotomies for evolution families, Commun. Pure Appl. Anal., 19 (2020), 1321-1336.  doi: 10.3934/cpaa.2020064.  Google Scholar

[18]

D. Dragičević, Admissibility and nonuniform polynomial dichotomies, Math. Nachr., 293 (2019), 226-243.  doi: 10.1002/mana.201800291.  Google Scholar

[19]

N. Lupa and M. Megan, Exponential dichotomies of evolution operators in Banach spaces, Monatsh. Math., 174 (2014), 265-284.  doi: 10.1007/s00605-013-0517-y.  Google Scholar

[20]

J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Academic, New York, 1966.  Google Scholar

[21]

Ng uyen Van MinhF. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line, Integral Equ. Oper. Theory, 32 (1998), 332-353.  doi: 10.1007/BF01203774.  Google Scholar

[22]

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

[23]

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

[24]

Y. Pesin, Families of invariant manifolds that corresponding to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976) 1332–1379, (Russian) English transl. Math. USSR-Izv. 10 (1976), 1261–1305.  Google Scholar

[25]

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russ. Math. Surv., 32 (1977), 55-114.   Google Scholar

[26]

Y. Pesin, Geodesic flows in closed Riemannian manifolds without focal points, Math. USSR-Izv., 11 (1977), 1195-1228.   Google Scholar

[27]

M. Pinto, Discrete dichotomies, Comput. Math. Appl., 28 (1994), 259-270.  doi: 10.1016/0898-1221(94)00114-6.  Google Scholar

[28]

P. PredaA. Pogan and C. Preda, $(L^p; L^q)$-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equ. Oper. Theory, 49 (2004), 405-418.  doi: 10.1007/s00020-002-1268-7.  Google Scholar

[29]

C. PredaP. Preda and A. Craciunescu, A version of a theorem of R. Datko for nonuniform exponential contractions, J. Math. Anal. Appl., 385 (2012), 572-581.  doi: 10.1016/j.jmaa.2011.06.082.  Google Scholar

[30]

C. PredaP. Preda and A. Craciunescu, Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations, J. Funct. Anal., 258 (2010), 729-757.  doi: 10.1016/j.jfa.2009.09.002.  Google Scholar

[31]

B. Sasu and A. L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 33 (2013), 3057-3084.  doi: 10.3934/dcds.2013.33.3057.  Google Scholar

[32]

A. L. SasuM. G. Babuţia 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

[33]

A. L. Sasu and B. Sasu, Admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line, Appl. Math. Comput., 378 (2020), 125167. doi: 10.1016/j. amc. 2020.125167.  Google Scholar

[34]

B. Sasu and A. L. Sasu, A. L. Sasu and B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces, Integral Equ. Oper. Theory, 54 (2006), 113-130.  doi: 10.1007/s00020-004-1347-z.  Google Scholar

[35]

B. Sasu and A. L. Sasu, Integral equations in the study of the asymptotic behavior of skew-product flows, Asymptotic Anal., 68 (2010), 135-153.   Google Scholar

[36]

L. ZhouK. Lu and W. Zhang, Equivalences between nonuniform exponential dichotomy and admissibility, J. Differ. Equ., 262 (2017), 682-747.  doi: 10.1016/j.jde.2016.09.035.  Google Scholar

[37]

L. Zhou and W. Zhang, Exponential dichotomy and $(\ell^p, \ell^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408.  doi: 10.1016/j.jmaa.2005.04.047.  Google Scholar

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