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Admissibility, a general type of Lipschitz shadowing and structural stability
1. | Department of Mathematics, University of Rijeka, 51000 Rijeka |
References:
[1] |
L. Barreira, D. Dragičević and C. Valls, Exponential dichotomies with respect to a sequence of norms and admissibility, Int. J. Math., 25, 1450024 (2014), 20 pages.
doi: 10.1142/S0129167X14500244. |
[2] |
L. Barreira, D. Dragičević and C. Valls, Nonuniform hyperbolicity and admissibility, Adv. Nonlinear Stud., 14 (2014), 791-811. |
[3] |
L. Barreira, D. Dragičević and C. Valls, Strong and weak $(L^p,L^q)$-admissibility, Bull. Sci. Math., 138 (2014), 721-741.
doi: 10.1016/j.bulsci.2013.11.005. |
[4] |
L. Barreira, D. Dragičević and C. Valls, Admissibility on the half line for evolution families, J. Anal. Math., to appear. |
[5] |
L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lect. Notes. in Math. 1926, 2008.
doi: 10.1007/978-3-540-74775-8. |
[6] |
A. Ben-Artzi and I. Gohberg, Dichotomy of systems and invertibility of linear ordinary differential operators, in Time-Variant Systems and Interpolation, Oper. Theory Adv. Appl., 56, Birkhäuser, 1992, pp. 90-119. |
[7] |
A. Ben-Artzi, I. Gohberg and M. Kaashoek, Invertibility and dichotomy of differential operators on a half-line, J. Dynam. Differential Equations, 5 (1993), 1-36.
doi: 10.1007/BF01063733. |
[8] |
C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs 70, Amer. Math. Soc., 1999.
doi: 10.1090/surv/070. |
[9] |
C. Coffman and J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166. |
[10] |
W. Coppel, Dichotomies in Stability Theory, Lect. Notes. in Math. 629, Springer, 1978. |
[11] |
Ju. Dalec'kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974. |
[12] |
D. Dragičević and S. Slijepčević, Characterization of hyperbolicity and generalized shadowing lemma, Dyn. Syst., 26 (2011), 483-502.
doi: 10.1080/14689367.2011.606205. |
[13] |
A. Fakhari, K. Lee, and K. Tajbakhsh, Diffeomorphisms with $L^p$-shadowing property, Acta Math. Sin., 27 (2011), 19-28.
doi: 10.1007/s10114-011-0050-7. |
[14] |
J. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, Amer. Math. Soc., 1988. |
[15] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math. 840, Springer, 1981. |
[16] |
N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354.
doi: 10.1016/j.jfa.2005.11.002. |
[17] |
Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414. |
[18] |
B. Levitan and V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, 1982. |
[19] |
A. D. Maizel, On stability of solutions of systems of differential equations, Trudi Uralskogo Politekhnicheskogo Instituta, Mathematics, 51 (1954), 20-50. |
[20] |
R. Ma né, 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. |
[21] |
J. Massera and J. Schäffer, Linear differential equations and functional analysis. I., Ann. of Math., 67 (1958), 517-573. |
[22] |
J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics 21, Academic Press, 1966. |
[23] |
N. Minh and N. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44.
doi: 10.1006/jmaa.2001.7450. |
[24] |
N. Van Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory, 32 (1998), 332-353.
doi: 10.1007/BF01203774. |
[25] |
K. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156.
doi: 10.2307/2047477. |
[26] |
K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer, Dordrecht, 2000.
doi: 10.1007/978-1-4757-3210-8. |
[27] |
K. J. Palmer, S. Yu. Pilyugin and S. B. Tikhmirov, Lipschitz shadowing and structural stability of flows, J. Differential Equations, 252 (2012), 1723-1747.
doi: 10.1016/j.jde.2011.07.026. |
[28] |
O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
[29] |
S. Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes Math., vol. 1706, Springer, Berlin, 1999. |
[30] |
S. Yu. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515.
doi: 10.1088/0951-7715/23/10/009. |
[31] |
S. Pilyugin, G. Volfson and D. Todorov, Dynamical systems with Lipschitz inverse shadowing properties, Vestnik St. Petersburg University: Mathematics, 44 (2011), 208-213.
doi: 10.3103/S106345411103006X. |
[32] |
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. |
[33] |
P. Preda, A. Pogan and C. Preda, $(L^p,L^q)$-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equations Operator Theory, 49 (2004), 405-418.
doi: 10.1007/s00020-002-1268-7. |
[34] |
A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920.
doi: 10.1016/j.jmaa.2008.03.019. |
[35] |
G. Sell and Y. You, Dynamics of Evolutionary Equation Applied Mathematical Sciences 143, Springer, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[36] |
S. Tikhomirov, Hölder shadowing on finite intervals, Ergodic Theory Dynam. Systems, (2014), http://dx.doi.org/10.1017/etds.2014.7 |
[37] |
D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in Shadowing Theory, Discrete Contin. Dyn. Syst., 33 (2013), 4187 - 4205.
doi: 10.3934/dcds.2013.33.4187. |
[38] |
W. Zhang, The Fredholm alternative and exponential dichotomies for parabolic equations, J. Math. Anal. Appl., 191 (1985), 180-201.
doi: 10.1016/S0022-247X(85)71126-2. |
show all references
References:
[1] |
L. Barreira, D. Dragičević and C. Valls, Exponential dichotomies with respect to a sequence of norms and admissibility, Int. J. Math., 25, 1450024 (2014), 20 pages.
doi: 10.1142/S0129167X14500244. |
[2] |
L. Barreira, D. Dragičević and C. Valls, Nonuniform hyperbolicity and admissibility, Adv. Nonlinear Stud., 14 (2014), 791-811. |
[3] |
L. Barreira, D. Dragičević and C. Valls, Strong and weak $(L^p,L^q)$-admissibility, Bull. Sci. Math., 138 (2014), 721-741.
doi: 10.1016/j.bulsci.2013.11.005. |
[4] |
L. Barreira, D. Dragičević and C. Valls, Admissibility on the half line for evolution families, J. Anal. Math., to appear. |
[5] |
L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lect. Notes. in Math. 1926, 2008.
doi: 10.1007/978-3-540-74775-8. |
[6] |
A. Ben-Artzi and I. Gohberg, Dichotomy of systems and invertibility of linear ordinary differential operators, in Time-Variant Systems and Interpolation, Oper. Theory Adv. Appl., 56, Birkhäuser, 1992, pp. 90-119. |
[7] |
A. Ben-Artzi, I. Gohberg and M. Kaashoek, Invertibility and dichotomy of differential operators on a half-line, J. Dynam. Differential Equations, 5 (1993), 1-36.
doi: 10.1007/BF01063733. |
[8] |
C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs 70, Amer. Math. Soc., 1999.
doi: 10.1090/surv/070. |
[9] |
C. Coffman and J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166. |
[10] |
W. Coppel, Dichotomies in Stability Theory, Lect. Notes. in Math. 629, Springer, 1978. |
[11] |
Ju. Dalec'kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974. |
[12] |
D. Dragičević and S. Slijepčević, Characterization of hyperbolicity and generalized shadowing lemma, Dyn. Syst., 26 (2011), 483-502.
doi: 10.1080/14689367.2011.606205. |
[13] |
A. Fakhari, K. Lee, and K. Tajbakhsh, Diffeomorphisms with $L^p$-shadowing property, Acta Math. Sin., 27 (2011), 19-28.
doi: 10.1007/s10114-011-0050-7. |
[14] |
J. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, Amer. Math. Soc., 1988. |
[15] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math. 840, Springer, 1981. |
[16] |
N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354.
doi: 10.1016/j.jfa.2005.11.002. |
[17] |
Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414. |
[18] |
B. Levitan and V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, 1982. |
[19] |
A. D. Maizel, On stability of solutions of systems of differential equations, Trudi Uralskogo Politekhnicheskogo Instituta, Mathematics, 51 (1954), 20-50. |
[20] |
R. Ma né, 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. |
[21] |
J. Massera and J. Schäffer, Linear differential equations and functional analysis. I., Ann. of Math., 67 (1958), 517-573. |
[22] |
J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics 21, Academic Press, 1966. |
[23] |
N. Minh and N. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44.
doi: 10.1006/jmaa.2001.7450. |
[24] |
N. Van Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory, 32 (1998), 332-353.
doi: 10.1007/BF01203774. |
[25] |
K. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156.
doi: 10.2307/2047477. |
[26] |
K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer, Dordrecht, 2000.
doi: 10.1007/978-1-4757-3210-8. |
[27] |
K. J. Palmer, S. Yu. Pilyugin and S. B. Tikhmirov, Lipschitz shadowing and structural stability of flows, J. Differential Equations, 252 (2012), 1723-1747.
doi: 10.1016/j.jde.2011.07.026. |
[28] |
O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
[29] |
S. Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes Math., vol. 1706, Springer, Berlin, 1999. |
[30] |
S. Yu. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515.
doi: 10.1088/0951-7715/23/10/009. |
[31] |
S. Pilyugin, G. Volfson and D. Todorov, Dynamical systems with Lipschitz inverse shadowing properties, Vestnik St. Petersburg University: Mathematics, 44 (2011), 208-213.
doi: 10.3103/S106345411103006X. |
[32] |
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. |
[33] |
P. Preda, A. Pogan and C. Preda, $(L^p,L^q)$-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equations Operator Theory, 49 (2004), 405-418.
doi: 10.1007/s00020-002-1268-7. |
[34] |
A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920.
doi: 10.1016/j.jmaa.2008.03.019. |
[35] |
G. Sell and Y. You, Dynamics of Evolutionary Equation Applied Mathematical Sciences 143, Springer, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[36] |
S. Tikhomirov, Hölder shadowing on finite intervals, Ergodic Theory Dynam. Systems, (2014), http://dx.doi.org/10.1017/etds.2014.7 |
[37] |
D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in Shadowing Theory, Discrete Contin. Dyn. Syst., 33 (2013), 4187 - 4205.
doi: 10.3934/dcds.2013.33.4187. |
[38] |
W. Zhang, The Fredholm alternative and exponential dichotomies for parabolic equations, J. Math. Anal. Appl., 191 (1985), 180-201.
doi: 10.1016/S0022-247X(85)71126-2. |
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