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Robustness of nonuniform exponential trichotomies in Banach spaces. (English) Zbl 1202.34105

Summary: We establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations
\[ x'=A(t)x \]
in Banach spaces. We also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy. We consider both trichotomies in semi-infinite intervals and trichotomies in \(\mathbb R\).

MSC:

34G10 Linear differential equations in abstract spaces
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
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[1] Barreira, L.; Pesin, Ya., Nonuniform Hyperbolicity, Encyclopedia Math. Appl., vol. 115 (2007), Cambridge University Press · Zbl 1144.37002
[2] Barreira, L.; Valls, C., A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential Equations, 228, 285-310 (2006) · Zbl 1099.37022
[3] Barreira, L.; Valls, C., Conjugacies for linear and nonlinear perturbations of nonuniform behavior, J. Funct. Anal., 253, 324-358 (2007) · Zbl 1127.37030
[4] Barreira, L.; Valls, C., Smooth center manifolds for nonuniformly partially hyperbolic trajectories, J. Differential Equations, 237, 307-342 (2007) · Zbl 1114.34038
[5] Barreira, L.; Valls, C., Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations, 244, 2407-2447 (2008) · Zbl 1158.34041
[6] Barreira, L.; Valls, C., Stability of Nonautonomous Differential Equations, Lecture Notes in Math., vol. 1926 (2008) · Zbl 1152.34003
[7] Carr, J., Applications of Centre Manifold Theory, Appl. Math. Sci., vol. 35 (1981), Springer · Zbl 0464.58001
[8] Chicone, C.; Latushkin, Yu., Center manifolds for infinite dimensional nonautonomous differential equations, J. Differential Equations, 141, 356-399 (1997) · Zbl 0992.34033
[9] Chow, S.-N.; Leiva, H., Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations, 120, 429-477 (1995) · Zbl 0831.34067
[10] Chow, S.-N.; Liu, W.; Yi, Y., Center manifolds for invariant sets, J. Differential Equations, 168, 355-385 (2000) · Zbl 0972.34033
[11] Chow, S.-N.; Liu, W.; Yi, Y., Center manifolds for smooth invariant manifolds, Trans. Amer. Math. Soc., 352, 5179-5211 (2000) · Zbl 0953.34038
[12] Coppel, W., Dichotomies and reducibility, J. Differential Equations, 3, 500-521 (1967) · Zbl 0162.39104
[13] Dalec’kiĭ, Ju.; Kreĭn, M., Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monogr., vol. 43 (1974), Amer. Math. Soc.
[14] Kelley, A., The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3, 546-570 (1967) · Zbl 0173.11001
[15] Massera, J.; Schäffer, J., Linear differential equations and functional analysis. I, Ann. of Math. (2), 67, 517-573 (1958) · Zbl 0178.17701
[16] Massera, J.; Schäffer, J., Linear Differential Equations and Function Spaces, Pure Appl. Math., vol. 21 (1966), Academic Press · Zbl 0202.14701
[17] Mielke, A., A reduction principle for nonautonomous systems in infinite-dimensional spaces, J. Differential Equations, 65, 68-88 (1986) · Zbl 0601.35018
[18] Naulin, R.; Pinto, M., Admissible perturbations of exponential dichotomy roughness, Nonlinear Anal., 31, 559-571 (1998) · Zbl 0902.34041
[19] Palmer, K., Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55, 225-256 (1984) · Zbl 0508.58035
[20] Perron, O., Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32, 703-728 (1930) · JFM 56.1040.01
[21] Pliss, V., A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat., 28, 1297-1324 (1964) · Zbl 0131.31505
[22] Pliss, V.; Sell, G., Robustness of exponential dichotomies in infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 11, 471-513 (1999) · Zbl 0941.37052
[23] Popescu, L., Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl., 314, 436-454 (2006) · Zbl 1093.34022
[24] Vanderbauwhede, A., Centre manifolds, normal forms and elementary bifurcations, (Dynamics Reported, vol. 2 (1989), Wiley), 89-169 · Zbl 0677.58001
[25] Vanderbauwhede, A.; Iooss, G., Center manifold theory in infinite dimensions, (Dynamics Reported (N.S.), vol. 1 (1992), Springer), 125-163 · Zbl 0751.58025
[26] Vanderbauwhede, A.; van Gils, S., Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal., 72, 209-224 (1987) · Zbl 0621.47050
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