American Institute of Mathematical Sciences

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

 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

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.
Citation: Dmitry Todorov. Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. doi: 10.1070/IM2011v075n04ABEH002548. [4] W. A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Mathematics, (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. 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. [8] Yu. Latushkin and Yu. Tomilov, Fredholm properties of evolution semigroups,, Illinois Journal of Mathematics, 48 (2004), 999. [9] R. Mañé, Characterizations of as diffeomorphisms,, in, (1977), 389. [10] A. D. Maĭzel, On stability of solutions of systems of differential equations, (Russian), Ural. Politehn. Inst. Trudy, 51 (1954), 20. [11] M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families,, Discrete Contin. Dyn. Syst., 9 (2003), 383. [12] K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,", Mathematics and its Applications, 501 (2000). [13] O. Perron, Die stabilittsfrage bei differentialgleichungen, (German), Mathematische Zeitschrift, 32 (1930), 703. 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. doi: 10.3103/S106345411103006X. [15] S. Yu. Pilyugin, "Shadowing in Dynamical Systems,", Lecture Notes in Mathematics, 1706 (1999). [16] _______, Generalizations of the notion of hyperbolicity,, Journal of Difference Equations and Applications, 12 (2006), 271. doi: 10.1080/10236190500489350. [17] _______, Sets of dynamical systems with various limit shadowing properties,, Journal of Dynamics and Differential Equations, 19 (2007), 747. doi: 10.1007/s10884-007-9073-2. [18] S. Yu Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability,, Nonlinearity, 23 (2010), 2509. doi: 10.1088/0951-7715/23/10/009. [19] V. A. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations,, in, (1977), 168. [20] K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds,, Osaka J. Math., 31 (1994), 373. [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. 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. doi: 10.1016/j.jmaa.2005.04.047. [24] Sergey Tikhomirov, Hölder shadowing and structural stability,, preprint, ().

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References:
 [1] A. G. Baskakov, On the invertibility and the Fredholm property of difference operators,, Mat. Zametki, 67 (2000), 816. 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. 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. doi: 10.1070/IM2011v075n04ABEH002548. [4] W. A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Mathematics, (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. 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. [8] Yu. Latushkin and Yu. Tomilov, Fredholm properties of evolution semigroups,, Illinois Journal of Mathematics, 48 (2004), 999. [9] R. Mañé, Characterizations of as diffeomorphisms,, in, (1977), 389. [10] A. D. Maĭzel, On stability of solutions of systems of differential equations, (Russian), Ural. Politehn. Inst. Trudy, 51 (1954), 20. [11] M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families,, Discrete Contin. Dyn. Syst., 9 (2003), 383. [12] K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,", Mathematics and its Applications, 501 (2000). [13] O. Perron, Die stabilittsfrage bei differentialgleichungen, (German), Mathematische Zeitschrift, 32 (1930), 703. 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. doi: 10.3103/S106345411103006X. [15] S. Yu. Pilyugin, "Shadowing in Dynamical Systems,", Lecture Notes in Mathematics, 1706 (1999). [16] _______, Generalizations of the notion of hyperbolicity,, Journal of Difference Equations and Applications, 12 (2006), 271. doi: 10.1080/10236190500489350. [17] _______, Sets of dynamical systems with various limit shadowing properties,, Journal of Dynamics and Differential Equations, 19 (2007), 747. doi: 10.1007/s10884-007-9073-2. [18] S. Yu Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability,, Nonlinearity, 23 (2010), 2509. doi: 10.1088/0951-7715/23/10/009. [19] V. A. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations,, in, (1977), 168. [20] K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds,, Osaka J. Math., 31 (1994), 373. [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. 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. doi: 10.1016/j.jmaa.2005.04.047. [24] Sergey Tikhomirov, Hölder shadowing and structural stability,, preprint, ().
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