Asymptotic stability of nonuniform behaviour

Dragičević, Davor; Zhang, Weinian

doi:10.1090/proc/14444

Asymptotic stability of nonuniform behaviour
HTML articles powered by AMS MathViewer

by Davor Dragičević and Weinian Zhang PDF

Proc. Amer. Math. Soc. 147 (2019), 2437-2451 Request permission

Abstract:

This paper is devoted to exponential dichotomies of nonautono- mous difference equations. Under the assumptions that $(A_m)_{m\in \mathbb {Z}}$ is a sequence of bounded operators acting on an arbitrary Banach space $X$ that admits a uniform exponential dichotomy and that $(B_m)_{m\in \mathbb {Z}}$ is a sequence of compact operators such that $\lim _{\lvert m\rvert \to \infty } \lVert B_m\rVert =0$, D. Henry proved that either the sequence $(A_m+B_m)_{m\in \mathbb {Z}}$ admits a uniform exponential dichotomy or there exists a bounded nonzero sequence $(x_m)_{m\in \mathbb {Z}}\subset X$ such that $x_{m+1}=(A_m+B_m)x_m$ for each $m\in \mathbb {Z}$. In this paper we prove Henry’s result in the setting of nonuniform exponential dichotomies. Then we obtain a result on roughness of the nonuniform exponential dichotomy and give stability of Lyapunov exponents. In addition, we establish corresponding results for dynamics with continuous time.

References

Wolfgang Arendt, Frank Räbiger, and Ahmed Sourour, Spectral properties of the operator equation $AX+XB=Y$, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 178, 133–149. MR 1280689, DOI 10.1093/qmath/45.2.133
Luis Barreira, Davor Dragičević, and Claudia Valls, Strong and weak $(L^p,L^q)$-admissibility, Bull. Sci. Math. 138 (2014), no. 6, 721–741. MR 3251453, DOI 10.1016/j.bulsci.2013.11.005
Luis Barreira, Davor Dragičević, and Claudia Valls, Exponential dichotomies with respect to a sequence of norms and admissibility, Internat. J. Math. 25 (2014), no. 3, 1450024, 20. MR 3189781, DOI 10.1142/S0129167X14500244
Luis Barreira, Davor Dragičević, and Claudia Valls, From one-sided dichotomies to two-sided dichotomies, Discrete Contin. Dyn. Syst. 35 (2015), no. 7, 2817–2844. MR 3343543, DOI 10.3934/dcds.2015.35.2817
Luis Barreira and Yakov B. Pesin, Lyapunov exponents and smooth ergodic theory, University Lecture Series, vol. 23, American Mathematical Society, Providence, RI, 2002. MR 1862379, DOI 10.1090/ulect/023
Luis Barreira and Claudia Valls, Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations 244 (2008), no. 10, 2407–2447. MR 2414396, DOI 10.1016/j.jde.2008.02.028
L. Barreira, C. Silva, and C. Valls, Nonuniform behavior and robustness, J. Differential Equations 246 (2009), no. 9, 3579–3608. MR 2515169, DOI 10.1016/j.jde.2008.10.009
Luis Barreira and Claudia Valls, Stability of nonautonomous differential equations, Lecture Notes in Mathematics, vol. 1926, Springer, Berlin, 2008. MR 2368551, DOI 10.1007/978-3-540-74775-8
Luis Barreira and Claudia Valls, Robustness of discrete dynamics via Lyapunov sequences, Comm. Math. Phys. 290 (2009), no. 1, 219–238. MR 2520512, DOI 10.1007/s00220-009-0762-z
A. Bento and C. Silva, Robustness of discrete nonuniform dichotomic behavior, preprint, https://arxiv.org/abs/1308.6820, 2013.
Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs, vol. 70, American Mathematical Society, Providence, RI, 1999. MR 1707332, DOI 10.1090/surv/070
Shui-Nee Chow and Hugo Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations 120 (1995), no. 2, 429–477. MR 1347351, DOI 10.1006/jdeq.1995.1117
Jifeng Chu, Robustness of nonuniform behavior for discrete dynamics, Bull. Sci. Math. 137 (2013), no. 8, 1031–1047. MR 3130345, DOI 10.1016/j.bulsci.2013.03.003
W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Company, Boston, Mass., 1965. MR 0190463
Ju. L. Dalec′kiĭ and M. G. Kreĭn, Stability of solutions of differential equations in Banach space, Translations of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by S. Smith. MR 0352639
Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
Daniel B. Henry, Exponential dichotomies, the shadowing lemma and homoclinic orbits in Banach spaces, Resenhas 1 (1994), no. 4, 381–401. Dynamical phase transitions (São Paulo, 1994). MR 1357942
Nguyen Thieu Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal. 235 (2006), no. 1, 330–354. MR 2216449, DOI 10.1016/j.jfa.2005.11.002
O. Leontiev and P. Feketa, A new criterion for the roughness of exponential dichotomy on $\Bbb R$, Miskolc Math. Notes 16 (2015), no. 2, 987–994. MR 3454157, DOI 10.18514/MMN.2015.1516
José Luis Massera and Juan Jorge Schäffer, Linear differential equations and function spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York-London, 1966. MR 0212324
Raúl Naulin and Manuel Pinto, Admissible perturbations of exponential dichotomy roughness, Nonlinear Anal. 31 (1998), no. 5-6, 559–571. MR 1487846, DOI 10.1016/S0362-546X(97)00423-9
Kenneth J. Palmer, A perturbation theorem for exponential dichotomies, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 25–37. MR 899938, DOI 10.1017/S0308210500018175
Oskar Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), no. 1, 703–728 (German). MR 1545194, DOI 10.1007/BF01194662
Victor A. Pliss and George R. Sell, Robustness of exponential dichotomies in infinite-dimensional dynamical systems, J. Dynam. Differential Equations 11 (1999), no. 3, 471–513. MR 1693858, DOI 10.1023/A:1021913903923
Liviu Horia Popescu, Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl. 314 (2006), no. 2, 436–454. MR 2185241, DOI 10.1016/j.jmaa.2005.04.011
Liviu Horia Popescu, Exponential dichotomy roughness and structural stability for evolution families without bounded growth and decay, Nonlinear Anal. 71 (2009), no. 3-4, 935–947. MR 2527514, DOI 10.1016/j.na.2008.11.009
Christian Pötzsche, Corrigendum on: A note on the dichotomy spectrum [MR2553938], J. Difference Equ. Appl. 18 (2012), no. 7, 1257–1261. MR 2946334, DOI 10.1080/10236198.2011.573790
Adina Luminiţa Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl. 344 (2008), no. 2, 906–920. MR 2426319, DOI 10.1016/j.jmaa.2008.03.019
George R. Sell and Yuncheng You, Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143, Springer-Verlag, New York, 2002. MR 1873467, DOI 10.1007/978-1-4757-5037-9
Linfeng Zhou, Kening Lu, and Weinian Zhang, Roughness of tempered exponential dichotomies for infinite-dimensional random difference equations, J. Differential Equations 254 (2013), no. 9, 4024–4046. MR 3029143, DOI 10.1016/j.jde.2013.02.007