×

Translation invariant spaces and asymptotic properties of variational equations. (English) Zbl 1229.49007

Summary: We present a new perspective concerning the study of the asymptotic behavior of variational equations by employing function spaces techniques. We give a complete description of the dichotomous behavior of the most general case of skew-product flows, without any assumption concerning the flow, the cocycle or the splitting of the state space, our study being based only on the solvability of some associated control systems between certain function spaces. The main results do not only point out new necessary and sufficient conditions for the existence of uniform and exponential dichotomy of skew-product flows, but also provide a clear chart of the connections between the classes of translation invariant function spaces that play the role of the input or output classes with respect to certain control systems. Finally, we emphasize the significance of each underlying hypothesis by illustrative examples and present several interesting applications.

MSC:

49J27 Existence theories for problems in abstract spaces
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] E. Braverman and S. Zhukovskiy, “The problem of a lazy tester, or exponential dichotomy for impulsive differential equations revisited,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 3, pp. 971-979, 2008. · Zbl 1217.34091 · doi:10.1016/j.nahs.2008.04.002
[2] C. Cuevas and C. Vidal, “Discrete dichotomies and asymptotic behavior for abstract retarded functional difference equations in phase space,” Journal of Difference Equations and Applications, vol. 8, no. 7, pp. 603-640, 2002. · Zbl 1019.39008 · doi:10.1080/10236190290032499
[3] S.-N. Chow and H. Leiva, “Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces,” Journal of Differential Equations, vol. 120, no. 2, pp. 429-477, 1995. · Zbl 0831.34067 · doi:10.1006/jdeq.1995.1117
[4] S.-N. Chow and H. Leiva, “Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces,” Proceedings of the American Mathematical Society, vol. 124, no. 4, pp. 1071-1081, 1996. · Zbl 0845.34064 · doi:10.1090/S0002-9939-96-03433-8
[5] S.-N. Chow and H. Leiva, “Unbounded perturbation of the exponential dichotomy for evolution equations,” Journal of Differential Equations, vol. 129, no. 2, pp. 509-531, 1996. · Zbl 0857.34062 · doi:10.1006/jdeq.1996.0125
[6] J. H. Liu, G. M. N’Guérékata, and N. Van Minh, Topics on Stability and Periodicity in Abstract Differential Equations, vol. 6 of Series on Concrete and Applicable Mathematics, World Scientific, Hackensack, NJ, USA, 2008. · Zbl 1158.34002 · doi:10.1142/9789812818249
[7] N. Van Minh, F. Räbiger, and R. Schnaubelt, “Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line,” Integral Equations and Operator Theory, vol. 32, no. 3, pp. 332-353, 1998. · Zbl 0977.34056 · doi:10.1007/BF01203774
[8] N. Van Minh, G. M. N’Guérékata, and R. Yuan, Lectures on the Asymptotic Behavior of Solutions of Differential Equations, Nova Science, New York, NY, USA, 2008. · Zbl 1162.34300
[9] K. J. Palmer, “Exponential dichotomies and Fredholm operators,” Proceedings of the American Mathematical Society, vol. 104, no. 1, pp. 149-156, 1988. · Zbl 0675.34006 · doi:10.2307/2047477
[10] K. J. Palmer, Shadowing in Dynamical Systems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. · Zbl 0997.37001
[11] O. Perron, “Die stabilitätsfrage bei differentialgleichungen,” Mathematische Zeitschrift, vol. 32, no. 1, pp. 703-728, 1930. · doi:10.1007/BF01194662
[12] M. Pituk, “A criterion for the exponential stability of linear difference equations,” Applied Mathematics Letters, vol. 17, no. 7, pp. 779-783, 2004. · Zbl 1068.39019 · doi:10.1016/j.aml.2004.06.005
[13] H. M. Rodrigues and J. G. Ruas-Filho, “Evolution equations: dichotomies and the Fredholm alternative for bounded solutions,” Journal of Differential Equations, vol. 119, no. 2, pp. 263-283, 1995. · Zbl 0837.34065 · doi:10.1006/jdeq.1995.1091
[14] B. Sasu and A. L. Sasu, “Input-output conditions for the asymptotic behavior of linear skew-product flows and applications,” Communications on Pure and Applied Analysis, vol. 5, no. 3, pp. 551-569, 2006. · Zbl 1142.47025 · doi:10.3934/cpaa.2006.5.551
[15] B. Sasu, “New criteria for exponential expansiveness of variational difference equations,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 287-297, 2007. · Zbl 1115.39005 · doi:10.1016/j.jmaa.2006.04.024
[16] B. Sasu, “Robust stability and stability radius for variational control systems,” Abstract and Applied Analysis, vol. 2008, Article ID 381791, 29 pages, 2008. · Zbl 1395.93469
[17] B. Sasu, “On dichotomous behavior of variational difference equations and applications,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 140369, 16 pages, 2009. · Zbl 1178.39028 · doi:10.1155/2009/140369
[18] B. Sasu, “Integral conditions for exponential dichotomy: a nonlinear approach,” Bulletin des Sciences Mathématiques, vol. 134, no. 3, pp. 235-246, 2010. · Zbl 1196.34066 · doi:10.1016/j.bulsci.2009.06.006
[19] C. Foias, G. R. Sell, and R. Temam, “Inertial manifolds for nonlinear evolutionary equations,” Journal of Differential Equations, vol. 73, no. 2, pp. 309-353, 1988. · Zbl 0643.58004 · doi:10.1016/0022-0396(88)90110-6
[20] W. A. Coppel, Dichotomies in Stability Theory, vol. 629 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1978. · Zbl 0376.34001
[21] J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, American Mathematical Society, Providence, RI, USA, 1974.
[22] J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, vol. 21 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1966. · Zbl 0243.34107
[23] L. Berezansky and E. Braverman, “On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations,” Journal of Mathematical Analysis and Applications, vol. 304, no. 2, pp. 511-530, 2005. · Zbl 1068.39004 · doi:10.1016/j.jmaa.2004.09.042
[24] A. D. Maĭzel’, “On stability of solutions of systems of differential equations,” Trudy Ural’skogo Politekhnicheskogo Instituta, vol. 51, pp. 20-50, 1954 (Russian).
[25] P. Meyer-Nieberg, Banach Lattices, Universitext, Springer, Berlin, Germany, 1991. · Zbl 0743.46015
[26] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, NY, USA, 3rd edition, 1987. · Zbl 0925.00005
[27] C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Boston, Mass, USA, 1988. · Zbl 0647.46057
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.