Admissibility and nonuniform exponential dichotomies for difference equations without bounded growth or Lyapunov norms

Wu, Mengda; Xia, Yonghui

doi:10.1090/proc/16485

Admissibility and nonuniform exponential dichotomies for difference equations without bounded growth or Lyapunov norms
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by Mengda Wu and Yonghui Xia;

Proc. Amer. Math. Soc. 151 (2023), 4389-4403

DOI: https://doi.org/10.1090/proc/16485

Published electronically: May 19, 2023

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Abstract:

Many previous works used the admissibility of function classes to characterize nonuniform exponential dichotomy (for short, NEDs) by employing Lyapunov norms. Recently, L. Zhou and W. Zhang [J. Funct. Anal. 271 (2016), pp. 1087–1129] characterized NEDs without Lyapunov norms. They utilized two admissible pairs of function classes and an assumption on certain subspaces to describe NEDs under the prerequisite of bounded growth, which plays an essential role in their arguments. However, in this paper, we remove the condition of bounded growth when characterizing NEDs. Neither bounded growth nor Lyapunov norms are used to describe NEDs in this paper.

References

Luis Barreira, Davor Dragičević, and Claudia Valls, Admissibility on the half line for evolution families, J. Anal. Math. 132 (2017), 157–176. MR 3666809, DOI 10.1007/s11854-017-0017-4
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 and Claudia Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential Equations 228 (2006), no. 1, 285–310. MR 2254432, DOI 10.1016/j.jde.2006.04.001
Luis Barreira and Claudia Valls, Nonuniform exponential dichotomies and admissibility, Discrete Contin. Dyn. Syst. 30 (2011), no. 1, 39–53. MR 2773131, DOI 10.3934/dcds.2011.30.39
Shui-Nee Chow and Kening Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations 74 (1988), no. 2, 285–317. MR 952900, DOI 10.1016/0022-0396(88)90007-1
Charles V. Coffman and Juan Jorge Schäffer, Dichotomies for linear difference equations, Math. Ann. 172 (1967), 139–166. MR 214946, DOI 10.1007/BF01350095
W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR 481196
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, RI, 1974. Translated from the Russian by S. Smith. MR 352639
Davor Dragičević, Weinian Zhang, and Linfeng Zhou, Admissibility and nonuniform exponential dichotomies, J. Differential Equations 326 (2022), 201–226. MR 4411608, DOI 10.1016/j.jde.2022.04.014
D. Dragičević, L. Zhou, and W. Zhang, Description of tempered exponential dichotomies by admissibility with no Lyapunov norms, Preprint, arXiv:2206.06778, 2022.
Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. MR 419901
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
Nguyen Thieu Huy and Nguyen Van Minh, Exponential dichotomy of difference equations and applications to evolution equations on the half-line, Comput. Math. Appl. 42 (2001), no. 3-5, 301–311. Advances in difference equations, III. MR 1837992, DOI 10.1016/S0898-1221(01)00155-9
Y. Latushkin, T. Randolph, and R. Schnaubelt, Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces, J. Dynam. Differential Equations 10 (1998), no. 3, 489–510. MR 1646630, DOI 10.1023/A:1022609414870
Yuri Latushkin and Yuri Tomilov, Fredholm differential operators with unbounded coefficients, J. Differential Equations 208 (2005), no. 2, 388–429. MR 2109561, DOI 10.1016/j.jde.2003.10.018
Ta Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math. 63 (1934), no. 1, 99–141 (German). MR 1555392, DOI 10.1007/BF02547352
A. D. Maĭzel′, On stability of solutions of systems of differential equations, Ural. Politehn. Inst. Trudy 51 (1954), 20–50 (Russian). MR 75384
J. L. Massera and J. J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math. (2) 67 (1958), 517–573. MR 96985, DOI 10.2307/1969871
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 212324
Mihail Megan, Adina Luminiţa Sasu, and Bogdan Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst. 9 (2003), no. 2, 383–397. MR 1952381, DOI 10.3934/dcds.2003.9.383
Mihail Megan, Adina Luminiţa Sasu, and Bogdan Sasu, Uniform exponential dichotomy and admissibility for linear skew-product semiflows, Recent advances in operator theory, operator algebras, and their applications, Oper. Theory Adv. Appl., vol. 153, Birkhäuser, Basel, 2005, pp. 185–195. MR 2105476, DOI 10.1007/3-7643-7314-8_{1}1
Kenneth J. Palmer, A generalization of Hartman’s linearization theorem, J. Math. Anal. Appl. 41 (1973), 753–758. MR 330650, DOI 10.1016/0022-247X(73)90245-X
Kenneth J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc. 104 (1988), no. 1, 149–156. MR 958058, DOI 10.1090/S0002-9939-1988-0958058-1
Oskar Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), no. 1, 703–728 (German). MR 1545194, DOI 10.1007/BF01194662
Ciprian Preda, A discrete Perron-Ta Li type theorem for the dichotomy of evolution operators, J. Math. Anal. Appl. 332 (2007), no. 1, 727–734. MR 2319694, DOI 10.1016/j.jmaa.2006.10.056
Ciprian Preda, $(L^p(\Bbb R_+,X),L^q(\Bbb R_+,X))$-admissibility and exponential dichotomies of cocycles, J. Differential Equations 249 (2010), no. 3, 578–598. MR 2646041, DOI 10.1016/j.jde.2010.02.018
Petre Preda, Alin Pogan, and Ciprian Preda, Schäffer spaces and exponential dichotomy for evolutionary processes, J. Differential Equations 230 (2006), no. 1, 378–391. MR 2270558, DOI 10.1016/j.jde.2006.02.004
H. M. Rodrigues and J. G. Ruas-Filho, Evolution equations: dichotomies and the Fredholm alternative for bounded solutions, J. Differential Equations 119 (1995), no. 2, 263–283. MR 1340540, DOI 10.1006/jdeq.1995.1091
H. M. Rodrigues and M. Silveira, On the relationship between exponential dichotomies and the Fredholm alternative, J. Differential Equations 73 (1988), no. 1, 78–81. MR 938215, DOI 10.1016/0022-0396(88)90118-0
Robert J. Sacker and George R. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), no. 3, 320–358. MR 501182, DOI 10.1016/0022-0396(78)90057-8
Adina Luminiţa Sasu and Bogdan Sasu, Discrete admissibility, $l^p$-spaces and exponential dichotomy on the real line, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13 (2006), no. 5, 551–561. MR 2267963
Bogdan Sasu and Adina Luminiţa Sasu, Exponential dichotomy and $(l^p,l^q)$-admissibility on the half-line, J. Math. Anal. Appl. 316 (2006), no. 2, 397–408. MR 2206678, DOI 10.1016/j.jmaa.2005.04.047
Wei Nian Zhang, Generalized exponential dichotomies and invariant manifolds for differential equations, Adv. in Math. (China) 22 (1993), no. 1, 1–45 (English, with English and Chinese summaries). MR 1264672
Wei Nian Zhang, The Fredholm alternative and exponential dichotomies for parabolic equations, J. Math. Anal. Appl. 191 (1995), no. 1, 180–201. MR 1323770, DOI 10.1016/S0022-247X(85)71126-2
Weinian Zhang and Ian Stewart, Bounded solutions for non-autonomous parabolic equations, Dynam. Stability Systems 11 (1996), no. 2, 109–120. MR 1396891, DOI 10.1080/02681119608806219
Linfeng Zhou and Weinian Zhang, Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Funct. Anal. 271 (2016), no. 5, 1087–1129. MR 3522003, DOI 10.1016/j.jfa.2016.06.005
Linfeng Zhou, Kening Lu, and Weinian Zhang, Equivalences between nonuniform exponential dichotomy and admissibility, J. Differential Equations 262 (2017), no. 1, 682–747. MR 3567499, DOI 10.1016/j.jde.2016.09.035
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