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On the dichotomic behavior of discrete dynamical systems on the half-line
1. | Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Blvd. No. 4, 300223 Timişoara, Romania |
References:
[1] |
N. Apreutesei and V. Volpert, Solvability conditions for infinite systems of difference equations, J. Difference Equ. Appl., 15 (2009), 659-678.
doi: 10.1080/10236190802259824. |
[2] |
B. Aulbach and N. Van Minh, The concept of spectral dichotomy for linear difference equations II, J. Difference Equ. Appl., 2 (1996), 251-262.
doi: 10.1080/10236199608808060. |
[3] |
L. Barreira and C. Valls, Stability of dichotomies in difference equations with infinite delay, Nonlinear Analysis, 72 (2010), 881-893.
doi: 10.1016/j.na.2009.07.028. |
[4] |
L. Barreira and C. Valls, Nonuniform exponential dichotomies and admissibility, Discrete Contin. Dyn. Syst., 30 (2011), 39-53.
doi: 10.3934/dcds.2011.30.39. |
[5] |
L. Barreira and C. Valls, Stable manifolds with optimal regularity for difference equations, Discrete Contin. Dyn. Syst., 32 (2012), 1537-1555.
doi: 10.3934/dcds.2012.32.1537. |
[6] |
L. Barreira and C. Valls, Nonautonomous difference equations and a Perron-type theorem, Bull. Sci. Math., 136 (2012), 277-290.
doi: 10.1016/j.bulsci.2011.12.003. |
[7] |
C. Bennett and R. Sharpley, "Interpolation of Operators," Pure Appl. Math., 129, 1988. |
[8] |
L. Berezansky and E. Braverman, On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations, J. Math. Anal. Appl., 304 (2005), 511-530.
doi: 10.1016/j.jmaa.2004.09.042. |
[9] |
L. Berezansky and E. Braverman, New stability conditions for linear difference equations using Bohl-Perron type theorems, J. Difference Equ. Appl., 17 (2011), 657-675.
doi: 10.1080/10236190903146938. |
[10] |
E. Braverman and I. M. Karabash, Bohl-Perron-type stability theorems for linear difference equations with infinite delay, J. Difference Equ. Appl., 18 (2012), 909-939.
doi: 10.1080/10236198.2010.531276. |
[11] |
S. N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces, J. Differential Equations, 120 (1995), 429-477.
doi: 10.1006/jdeq.1995.1117. |
[12] |
S. N. Chow and H. Leiva, Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces, Proc. Amer. Math. Soc., 124 (1996), 1071-1081.
doi: 10.1090/S0002-9939-96-03433-8. |
[13] |
C. V. Coffman and J. J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166. |
[14] |
W. A. Coppel, "Dichotomies in Stability Theory," Springer Verlag, Berlin, Heidelberg, New-York, 1978. |
[15] |
S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Difference Equ. Appl., 3 (1998), 417-448.
doi: 10.1080/10236199708808113. |
[16] |
S. Elaydi, Is the world evolving discretely?, Adv. Appl. Math., 31 (2003), 1-9.
doi: 10.1016/S0196-8858(03)00072-1. |
[17] |
N. T. Huy and N. Van Minh, Exponential dichotomy of difference equations and applications to evolution equations on the half-line, Comput. Math. Appl., 42 (2001), 301-311.
doi: 10.1016/S0898-1221(01)00155-9. |
[18] |
J. L. Massera and J. J. Schäffer, "Linear Differential Equations and Function Spaces," Academic Press, 1966. |
[19] |
M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst., 9 (2003), 383-397. |
[20] |
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. |
[21] |
N. Van Minh and N. T. 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. |
[22] |
N. Van Minh, Asymptotic behavior of individual orbits of discrete systems, Proc. Amer. Math. Soc., 137 (2009), 3025-3035.
doi: 10.1090/S0002-9939-09-09871-2. |
[23] |
O. Perron, Die Stabilitätsfrage bei Differentialgleischungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
[24] |
M. Pituk, A criterion for the exponential stability of linear difference equations, Appl. Math. Lett., 17 (2004), 779-783.
doi: 10.1016/j.aml.2004.06.005. |
[25] |
C. Pötzsche, "Geometric Theory of Discrete Nonautonomous Dynamical Systems," Lecture Notes in Mathematics, 2002, Springer, 2010.
doi: 10.1007/978-3-642-14258-1. |
[26] |
B. Sasu and A. L. Sasu, Stability and stabilizability for linear systems of difference equations, J. Difference Equ. Appl., 10 (2004), 1085-1105.
doi: 10.1080/10236190412331314178. |
[27] |
A. L. Sasu and B. Sasu, Exponential dichotomy and admissibility for evolution families on the real line, Dynam. Contin. Discrete Impuls. Systems, 13 (2006), 1-26. |
[28] |
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. |
[29] |
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. |
[30] |
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. |
[31] |
B. Sasu, Stability of difference equations and applications to robustness problems, Adv. Difference Equ., (2010), Article ID 869608, 24 pp. |
[32] |
A. L. Sasu and B. Sasu, Integral equations, dichotomy of evolution families on the half-line and applications, Integral Equations Operator Theory, 66 (2010), 113-140
doi: 10.1007/s00020-009-1735-5. |
[33] |
A. L. Sasu and B. Sasu, Input-output admissibility and exponential trichotomy of difference equations, J. Math. Anal. Appl., 380 (2011), 17-32.
doi: 10.1016/j.jmaa.2011.02.045. |
show all references
References:
[1] |
N. Apreutesei and V. Volpert, Solvability conditions for infinite systems of difference equations, J. Difference Equ. Appl., 15 (2009), 659-678.
doi: 10.1080/10236190802259824. |
[2] |
B. Aulbach and N. Van Minh, The concept of spectral dichotomy for linear difference equations II, J. Difference Equ. Appl., 2 (1996), 251-262.
doi: 10.1080/10236199608808060. |
[3] |
L. Barreira and C. Valls, Stability of dichotomies in difference equations with infinite delay, Nonlinear Analysis, 72 (2010), 881-893.
doi: 10.1016/j.na.2009.07.028. |
[4] |
L. Barreira and C. Valls, Nonuniform exponential dichotomies and admissibility, Discrete Contin. Dyn. Syst., 30 (2011), 39-53.
doi: 10.3934/dcds.2011.30.39. |
[5] |
L. Barreira and C. Valls, Stable manifolds with optimal regularity for difference equations, Discrete Contin. Dyn. Syst., 32 (2012), 1537-1555.
doi: 10.3934/dcds.2012.32.1537. |
[6] |
L. Barreira and C. Valls, Nonautonomous difference equations and a Perron-type theorem, Bull. Sci. Math., 136 (2012), 277-290.
doi: 10.1016/j.bulsci.2011.12.003. |
[7] |
C. Bennett and R. Sharpley, "Interpolation of Operators," Pure Appl. Math., 129, 1988. |
[8] |
L. Berezansky and E. Braverman, On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations, J. Math. Anal. Appl., 304 (2005), 511-530.
doi: 10.1016/j.jmaa.2004.09.042. |
[9] |
L. Berezansky and E. Braverman, New stability conditions for linear difference equations using Bohl-Perron type theorems, J. Difference Equ. Appl., 17 (2011), 657-675.
doi: 10.1080/10236190903146938. |
[10] |
E. Braverman and I. M. Karabash, Bohl-Perron-type stability theorems for linear difference equations with infinite delay, J. Difference Equ. Appl., 18 (2012), 909-939.
doi: 10.1080/10236198.2010.531276. |
[11] |
S. N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces, J. Differential Equations, 120 (1995), 429-477.
doi: 10.1006/jdeq.1995.1117. |
[12] |
S. N. Chow and H. Leiva, Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces, Proc. Amer. Math. Soc., 124 (1996), 1071-1081.
doi: 10.1090/S0002-9939-96-03433-8. |
[13] |
C. V. Coffman and J. J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166. |
[14] |
W. A. Coppel, "Dichotomies in Stability Theory," Springer Verlag, Berlin, Heidelberg, New-York, 1978. |
[15] |
S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Difference Equ. Appl., 3 (1998), 417-448.
doi: 10.1080/10236199708808113. |
[16] |
S. Elaydi, Is the world evolving discretely?, Adv. Appl. Math., 31 (2003), 1-9.
doi: 10.1016/S0196-8858(03)00072-1. |
[17] |
N. T. Huy and N. Van Minh, Exponential dichotomy of difference equations and applications to evolution equations on the half-line, Comput. Math. Appl., 42 (2001), 301-311.
doi: 10.1016/S0898-1221(01)00155-9. |
[18] |
J. L. Massera and J. J. Schäffer, "Linear Differential Equations and Function Spaces," Academic Press, 1966. |
[19] |
M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst., 9 (2003), 383-397. |
[20] |
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. |
[21] |
N. Van Minh and N. T. 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. |
[22] |
N. Van Minh, Asymptotic behavior of individual orbits of discrete systems, Proc. Amer. Math. Soc., 137 (2009), 3025-3035.
doi: 10.1090/S0002-9939-09-09871-2. |
[23] |
O. Perron, Die Stabilitätsfrage bei Differentialgleischungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
[24] |
M. Pituk, A criterion for the exponential stability of linear difference equations, Appl. Math. Lett., 17 (2004), 779-783.
doi: 10.1016/j.aml.2004.06.005. |
[25] |
C. Pötzsche, "Geometric Theory of Discrete Nonautonomous Dynamical Systems," Lecture Notes in Mathematics, 2002, Springer, 2010.
doi: 10.1007/978-3-642-14258-1. |
[26] |
B. Sasu and A. L. Sasu, Stability and stabilizability for linear systems of difference equations, J. Difference Equ. Appl., 10 (2004), 1085-1105.
doi: 10.1080/10236190412331314178. |
[27] |
A. L. Sasu and B. Sasu, Exponential dichotomy and admissibility for evolution families on the real line, Dynam. Contin. Discrete Impuls. Systems, 13 (2006), 1-26. |
[28] |
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. |
[29] |
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. |
[30] |
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. |
[31] |
B. Sasu, Stability of difference equations and applications to robustness problems, Adv. Difference Equ., (2010), Article ID 869608, 24 pp. |
[32] |
A. L. Sasu and B. Sasu, Integral equations, dichotomy of evolution families on the half-line and applications, Integral Equations Operator Theory, 66 (2010), 113-140
doi: 10.1007/s00020-009-1735-5. |
[33] |
A. L. Sasu and B. Sasu, Input-output admissibility and exponential trichotomy of difference equations, J. Math. Anal. Appl., 380 (2011), 17-32.
doi: 10.1016/j.jmaa.2011.02.045. |
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