Exponential behavior in Banach spaces: robustness of trichotomies in discrete time. (English) Zbl 1363.39023
Summary: For difference equations in Banach spaces, we consider a generalization of the notion of exponential dichotomy, usually called trichotomy in the literature, for which the behaviors in \(\mathbb Z^+\) and \(\mathbb Z^-\) are still exponential but need not agree at the origin. Our main aim is to show that this exponential behavior is robust, in the sense that it persists under sufficiently small linear perturbations.
MSC:
| 39A30 | Stability theory for difference equations |
| 34D09 | Dichotomy, trichotomy of solutions to ordinary differential equations |
| 34D10 | Perturbations of ordinary differential equations |
References:
| [1] | A. Alonso, J. Hong, R. Obaya, Exponential dichotomy and trichotomy for difference equations. Comput. Math. Appl. 38, 41-49 (1999) · Zbl 0939.39003 · doi:10.1016/S0898-1221(99)00167-4 |
| [2] | L. Barreira, C. Silva, C. Valls, Nonuniform behavior and robustness. J. Differ. Equ. 246, 3579-3608 (2009) · Zbl 1172.37013 · doi:10.1016/j.jde.2008.10.009 |
| [3] | C. Chicone, Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, vol 70 (American Mathematical Society, 1999) · Zbl 0970.47027 |
| [4] | S.-N. Chow, H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces. J. Differ. Equ. 120, 429-477 (1995) · Zbl 0831.34067 · doi:10.1006/jdeq.1995.1117 |
| [5] | W. Coppel, Dichotomies and reducibility. J. Differ. Equ. 3, 500-521 (1967) · Zbl 0162.39104 · doi:10.1016/0022-0396(67)90014-9 |
| [6] | W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, vol 629 (Springer, Berlin, 1978) · Zbl 0376.34001 |
| [7] | Ju. Dalec’kiĭ, M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, vol 43 (American Mathematical Society, 1974) · Zbl 0687.39003 |
| [8] | J. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol 25 (American Mathematical Society, 1988) · Zbl 0642.58013 |
| [9] | D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Mathematics, vol 840 (Springer, Berlin, 1981) · Zbl 0456.35001 |
| [10] | J. Massera, J. Schäffer, Linear differential equations and functional analysis. I. Ann. Math. 67(2), 517-573 (1958) · Zbl 0178.17701 · doi:10.2307/1969871 |
| [11] | O. Méndez, L.H. Popescu, On admissible perturbations for exponential dichotomy. J. Math. Anal. Appl. 337, 425-430 (2008) · Zbl 1135.34324 · doi:10.1016/j.jmaa.2007.04.006 |
| [12] | R. Naulin, M. Pinto, Admissible perturbations of exponential dichotomy roughness. Nonlinear Anal. 31, 559-571 (1998) · Zbl 0902.34041 · doi:10.1016/S0362-546X(97)00423-9 |
| [13] | G. Papaschinopoulos, On exponential trichotomy of linear difference equations. Appl. Anal. 40, 89-109 (1991) · Zbl 0687.39003 · doi:10.1080/00036819108839996 |
| [14] | O. Perron, Die Stabilitätsfrage bei Differentialgleichungen. Math. Z. 32, 703-728 (1930) · JFM 56.1040.01 · doi:10.1007/BF01194662 |
| [15] | V. Pliss, G. Sell, Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dyn. Differ. Equ. 11, 471-513 (1999) · Zbl 0941.37052 · doi:10.1023/A:1021913903923 |
| [16] | L.H. Popescu, Exponential dichotomy roughness on Banach spaces. J. Math. Anal. Appl. 314, 436-454 (2006) · Zbl 1093.34022 · doi:10.1016/j.jmaa.2005.04.011 |
| [17] | L.H. Popescu, Exponential dichotomy roughness and structural stability for evolution families without bounded growth and decay. Nonlinear Anal. 71, 935-947 (2009) · Zbl 1175.34071 · doi:10.1016/j.na.2008.11.009 |
| [18] | G. Sell, Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, vol 143 (Springer, Berlin, 2002) · Zbl 1254.37002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
