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July  2013, 33(7): 3057-3084. doi: 10.3934/dcds.2013.33.3057

## 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

Received  March 2012 Revised  November 2012 Published  January 2013

The aim of this paper is to obtain new criteria for the existence of the dichotomies of dynamical systems on the half-line. We associate to a discrete dynamical system an input-output system between two abstract sequence spaces. We deduce conditions for the existence of ordinary dichotomy and exponential dichotomy of the initial discrete system, by using certain admissibility properties of the associated input-output system. We establish the axiomatic structures of the input and output spaces, in each case, clarifying the underlying hypotheses as well as the generality of the proposed method. Next, we present a new and direct proof for the equivalence between the exponential dichotomy of an evolution family on the half-line and the exponential dichotomy of the associated discrete dynamical system. Finally, we apply our main results to the study of the exponential dichotomy of evolution families on the half-line.
Citation: Bogdan Sasu, Adina Luminiţa Sasu. On the dichotomic behavior of discrete dynamical systems on the half-line. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3057-3084. doi: 10.3934/dcds.2013.33.3057
##### 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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##### 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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