We formulate new conditions of Barbashin type for exponential stability of linear cocycles on arbitrary Banach spaces. We consider both cocycles over maps and flows. Our arguments rely on ergodic theory.
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Abu Alhalawa, M., Dragičević, D.: New conditions for (non)uniform behaviour of linear cocycles over flows. J. Math. Anal. Appl. 473, 367–381 (2019)
Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics. Springer, Berlin (1998)
Barbashin, E.A.: Introduction to Stability Theory. Izd, Nauka (1967). (in Russian)
Bataran, F., Ponce, R., Preda, C.: Discrete-time theorems for global and pointwise dichotomies of cocycles over semiflows. Monatsh. Math. 186, 579–607 (2018)
Buse, C., Megan, M., Prajea, M.-S., Preda, P.: The strong variant of a Barbashin theorem on stability of solutions for non-autonomous differential equations in Banach spaces. Integr. Equ. Oper. Theory 59, 491–500 (2007)
Chicone, C., Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs 70. American Mathematical Society, Providence (1999)
Chow, S.N., Leiva, H.: Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces. J. Differ. Equ. 120, 429–477 (1995)
Dragičević, D.: A version of a theorem of R. Datko for stability in average. Syst. Control Lett. 96, 1–6 (2016)
Dragičević, D.: Datko–Pazy conditions for nonuniform exponential stability. J. Differ. Equ. Appl. 24, 344–357 (2018)
Hai, P.V.: Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows. Appl. Anal. 90, 1897–1907 (2011)
Latushkin, Y., Schnaubelt, R.: volution semigroups, translation algebra and exponential dichotomy of cocycles. J. Differ. Equ. 159, 321–369 (1999)
Kingman, J.F.C.: Sub-additive ergodic theory. Ann. Probab. 1, 883–909 (1973)
Mather, J.: Characterization of Anosov diffeomorphisms. Indag. Math. 30, 479–483 (1968)
Megan, M., Sasu, A.L., Sasu, B.: On uniform exponential stability of linear skew-product semiflows in Banach spaces. Bull. Belg. Math. Soc. Simon Stevin 9, 143–154 (2002)
Megan, M., Sasu, A.L., Sasu, B.: Theorems of Perron type for uniform exponential dichotomy of linear skew-product semiflows. Bull. Belg. Math. Soc. Simon Stevin 10, 1–21 (2003)
Pesin, Y.: Families of invariant manifolds corresponding to nonzero characteristic exponents. Math. USSR-Izv. 40, 1261–1305 (1976)
Pesin, Y.: Characteristic Ljapunov exponents, and smooth ergodic theory. Russ. Math. Surv. 32, 55–114 (1977)
Popa, I.-L., Ceasu, T., Megan, M.: On exponential stability for linear discrete-time systems in Banach spaces. Comput. Math. Appl. 63, 1497–1503 (2012)
Preda, C., Preda, P., Petre, A.: On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow. Commun. Pure Appl. Anal. 8, 1637–1645 (2009)
Preda, C.: \((L^p(_+, X), L^q(_+, X)\)-admissibility and exponential dichotomy for cocycles. J. Differ. Equ. 249, 578–598 (2010)
Preda, C., Preda, P., Craciunescu, A.: Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations. J. Funct. Anal. 258, 729–757 (2010)
Preda, C., Preda, P., Bǎtǎran, F.: An extension of a theorem of R. Datko to the case of (non)uniform exponential stability of linear skew-product semiflows. J. Math. Anal. Appl. 425, 1148–1154 (2015)
Preda, C., Onofrei, O.R.: Nonuniform exponential dichotomy for linear skew-product semiflows over semiflows. Semigroup Forum 96, 241–252 (2018)
Sasu, A.L., Sasu, B.: Exponential stability for linear skew-product flows. Bull. Sci. Math. 128, 727–738 (2004)
Sasu, A.L., Sasu, B.: Input-output conditions for the asymptotic behavior of linear skew-product flows and applications. Commun. Pure Appl. Anal. 5, 551–569 (2006)
Sasu, A.L., Sasu, B.: Admissibility and exponential trichotomy of dynamical systems described by skew-product flows. J. Differ. Equ. 260, 1656–1689 (2016)
Sasu, B.: Integral conditions for exponential dichotomy: a nonlinear approach. Bull. Sci. Math. 134, 235–246 (2010)
Stoica, C., Megan, M.: On uniform exponential stability for skew-evolution semiflows on Banach spaces. Nonlinear Anal. 72, 1305–1313 (2010)
Viana, M., Oliveira, K.: Foundations of Ergodic Theory. Cambridge Studies in Advance Mathematics. Cambridge University Press, Cambridge (2016)
Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin (1981)
Zhou, L., Lu, K., Zhang, W.: Roughness of tempered dichotomies for infinite-dimensional random difference equations. J. Differ. Equ. 254, 4024–4046 (2013)
I would like to thank the anonymous referees for their constructive comments that helped me to correct some inaccuracies in the first version of the paper.
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Davor Dragičević was supported in part by Croatian Science Foundation under the Project IP-2019-04-1239 and by the University of Rijeka under the Projects uniri-prirod-18-9 and uniri-prprirod-19-16, 18.104.22.168.01.
Communicated by H. Bruin.
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Dragičević, D. Barbashin-type conditions for exponential stability of linear cocycles. Monatsh Math (2020). https://doi.org/10.1007/s00605-020-01438-z
- Lyapunov exponents
- Barbashin-type theorem
Mathematics Subject Classification
- Primary 34D20