Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory

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September 2013, 33(9): 4187-4205. doi: 10.3934/dcds.2013.33.4187

Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory

Dmitry Todorov ^1,

1.	Chebyshev laboratory, Saint Petersburg State University, 14th line of Vasiljevsky Island, 29B, Saint-Petersburg, 199178, Russian Federation

Received February 2012 Revised February 2013 Published March 2013

Abstract
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We generalize two classical results of Maizel and Pliss that describe relations between hyperbolicity properties of linear system of difference equations and its ability to have a bounded solution for every bounded inhomogeneity. We also apply one of this generalizations in shadowing theory of diffeomorphisms to prove that some sort of limit shadowing is equivalent to structural stability.

Keywords: difference equations, limit shadowing, Admissibility, hyperbolicity, structural stability..

Mathematics Subject Classification: Primary: 34D09, 34K12; Secondary: 37C50, 34D3.

Citation: Dmitry Todorov. Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4187-4205. doi: 10.3934/dcds.2013.33.4187

References:

[1]	A. G. Baskakov, On the invertibility and the Fredholm property of difference operators, Mat. Zametki, 67 (2000), 816-827. doi: 10.1007/BF02675622.
[2]	_______, Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations, Izvestiya: Mathematics, 73 (2009), 215-278. doi: 10.1070/IM2009v073n02ABEH002445.
[3]	M. S. Bichegkuev, On conditions for invertibility of difference and differential operators in weight spaces, Izvestiya: Mathematics, 75 (2011), 665-680. doi: 10.1070/IM2011v075n04ABEH002548.
[4]	W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978.
[5]	Yu. L. Dalecki and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Spaces," Moscow, 1970.
[6]	A. Fakhari, K. Lee and K. Tajbakhsh, Diffeomorphisms with $L^p$-shadowing property, Acta Math. Sin. (Engl. Ser.), 27 (2011), 19-28. doi: 10.1007/s10114-011-0050-7.
[7]	Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, Journal of Operator Theory, 58 (2007), 387-414.
[8]	Yu. Latushkin and Yu. Tomilov, Fredholm properties of evolution semigroups, Illinois Journal of Mathematics, 48 (2004), 999-1020.
[9]	R. Mañé, Characterizations of as diffeomorphisms, in "Geometry and Topology" (eds. Jacob Palis and Manfredo do Carmo), Lecture Notes in Mathematics, Vol. 597, Springer Berlin, (1977), 389-394.
[10]	A. D. Maĭzel, On stability of solutions of systems of differential equations, (Russian) Ural. Politehn. Inst. Trudy, 51 (1954), 20-50.
[11]	M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst., 9 (2003), 383-397.
[12]	K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications," Mathematics and its Applications, 501, Kluwer Academic Publishers, Dordrecht, 2000.
[13]	O. Perron, Die stabilittsfrage bei differentialgleichungen, (German) Mathematische Zeitschrift, 32 (1930), 703-728. doi: 10.1007/BF01194662.
[14]	S. Pilyugin, G. Vol'fson and D. Todorov, Dynamical systems with Lipschitz inverse shadowing properties, Vestnik St. Petersburg University: Mathematics, 44 (2011), 208-213. doi: 10.3103/S106345411103006X.
[15]	S. Yu. Pilyugin, "Shadowing in Dynamical Systems," Lecture Notes in Mathematics, 1706, Springer-Verlag, Berlin, 1999.
[16]	_______, Generalizations of the notion of hyperbolicity, Journal of Difference Equations and Applications, 12 (2006), 271-282. doi: 10.1080/10236190500489350.
[17]	_______, Sets of dynamical systems with various limit shadowing properties, Journal of Dynamics and Differential Equations, 19 (2007), 747-775. doi: 10.1007/s10884-007-9073-2.
[18]	S. Yu Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009.
[19]	V. A. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations, in "Problems of Asymptotic Theory of Nonlinear Oscillations" (Russian), Naukova Dumka, Kiev, (1977), 168-173.
[20]	K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.
[21]	A. L. Sasu and B. Sasu, Translation invariant spaces and asymptotic properties of variational equations, Abstract and Applied Analysis, 2011 (2011). doi: 10.1155/2011/539026.
[22]	B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478. doi: 10.1016/j.jmaa.2005.12.002.
[23]	B. Sasu and A. L. Sasu, Exponential dichotomy and $(l^p,l^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047.
[24]	Sergey Tikhomirov, Hölder shadowing and structural stability, preprint, arXiv:1106.4053v1.

show all references

References:

[1]	A. G. Baskakov, On the invertibility and the Fredholm property of difference operators, Mat. Zametki, 67 (2000), 816-827. doi: 10.1007/BF02675622.
[2]	_______, Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations, Izvestiya: Mathematics, 73 (2009), 215-278. doi: 10.1070/IM2009v073n02ABEH002445.
[3]	M. S. Bichegkuev, On conditions for invertibility of difference and differential operators in weight spaces, Izvestiya: Mathematics, 75 (2011), 665-680. doi: 10.1070/IM2011v075n04ABEH002548.
[4]	W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978.
[5]	Yu. L. Dalecki and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Spaces," Moscow, 1970.
[6]	A. Fakhari, K. Lee and K. Tajbakhsh, Diffeomorphisms with $L^p$-shadowing property, Acta Math. Sin. (Engl. Ser.), 27 (2011), 19-28. doi: 10.1007/s10114-011-0050-7.
[7]	Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, Journal of Operator Theory, 58 (2007), 387-414.
[8]	Yu. Latushkin and Yu. Tomilov, Fredholm properties of evolution semigroups, Illinois Journal of Mathematics, 48 (2004), 999-1020.
[9]	R. Mañé, Characterizations of as diffeomorphisms, in "Geometry and Topology" (eds. Jacob Palis and Manfredo do Carmo), Lecture Notes in Mathematics, Vol. 597, Springer Berlin, (1977), 389-394.
[10]	A. D. Maĭzel, On stability of solutions of systems of differential equations, (Russian) Ural. Politehn. Inst. Trudy, 51 (1954), 20-50.
[11]	M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst., 9 (2003), 383-397.
[12]	K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications," Mathematics and its Applications, 501, Kluwer Academic Publishers, Dordrecht, 2000.
[13]	O. Perron, Die stabilittsfrage bei differentialgleichungen, (German) Mathematische Zeitschrift, 32 (1930), 703-728. doi: 10.1007/BF01194662.
[14]	S. Pilyugin, G. Vol'fson and D. Todorov, Dynamical systems with Lipschitz inverse shadowing properties, Vestnik St. Petersburg University: Mathematics, 44 (2011), 208-213. doi: 10.3103/S106345411103006X.
[15]	S. Yu. Pilyugin, "Shadowing in Dynamical Systems," Lecture Notes in Mathematics, 1706, Springer-Verlag, Berlin, 1999.
[16]	_______, Generalizations of the notion of hyperbolicity, Journal of Difference Equations and Applications, 12 (2006), 271-282. doi: 10.1080/10236190500489350.
[17]	_______, Sets of dynamical systems with various limit shadowing properties, Journal of Dynamics and Differential Equations, 19 (2007), 747-775. doi: 10.1007/s10884-007-9073-2.
[18]	S. Yu Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009.
[19]	V. A. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations, in "Problems of Asymptotic Theory of Nonlinear Oscillations" (Russian), Naukova Dumka, Kiev, (1977), 168-173.
[20]	K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.
[21]	A. L. Sasu and B. Sasu, Translation invariant spaces and asymptotic properties of variational equations, Abstract and Applied Analysis, 2011 (2011). doi: 10.1155/2011/539026.
[22]	B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478. doi: 10.1016/j.jmaa.2005.12.002.
[23]	B. Sasu and A. L. Sasu, Exponential dichotomy and $(l^p,l^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047.
[24]	Sergey Tikhomirov, Hölder shadowing and structural stability, preprint, arXiv:1106.4053v1.

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