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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 & Continuous Dynamical Systems, 2013, 33 (7) : 3057-3084. doi: 10.3934/dcds.2013.33.3057
References:
[1]

J. Difference Equ. Appl., 15 (2009), 659-678. doi: 10.1080/10236190802259824.  Google Scholar

[2]

J. Difference Equ. Appl., 2 (1996), 251-262. doi: 10.1080/10236199608808060.  Google Scholar

[3]

Nonlinear Analysis, 72 (2010), 881-893. doi: 10.1016/j.na.2009.07.028.  Google Scholar

[4]

Discrete Contin. Dyn. Syst., 30 (2011), 39-53. doi: 10.3934/dcds.2011.30.39.  Google Scholar

[5]

Discrete Contin. Dyn. Syst., 32 (2012), 1537-1555. doi: 10.3934/dcds.2012.32.1537.  Google Scholar

[6]

Bull. Sci. Math., 136 (2012), 277-290. doi: 10.1016/j.bulsci.2011.12.003.  Google Scholar

[7]

Pure Appl. Math., 129, 1988.  Google Scholar

[8]

J. Math. Anal. Appl., 304 (2005), 511-530. doi: 10.1016/j.jmaa.2004.09.042.  Google Scholar

[9]

J. Difference Equ. Appl., 17 (2011), 657-675. doi: 10.1080/10236190903146938.  Google Scholar

[10]

J. Difference Equ. Appl., 18 (2012), 909-939. doi: 10.1080/10236198.2010.531276.  Google Scholar

[11]

J. Differential Equations, 120 (1995), 429-477. doi: 10.1006/jdeq.1995.1117.  Google Scholar

[12]

Proc. Amer. Math. Soc., 124 (1996), 1071-1081. doi: 10.1090/S0002-9939-96-03433-8.  Google Scholar

[13]

Math. Ann., 172 (1967), 139-166.  Google Scholar

[14]

Springer Verlag, Berlin, Heidelberg, New-York, 1978.  Google Scholar

[15]

J. Difference Equ. Appl., 3 (1998), 417-448. doi: 10.1080/10236199708808113.  Google Scholar

[16]

Adv. Appl. Math., 31 (2003), 1-9. doi: 10.1016/S0196-8858(03)00072-1.  Google Scholar

[17]

Comput. Math. Appl., 42 (2001), 301-311. doi: 10.1016/S0898-1221(01)00155-9.  Google Scholar

[18]

Academic Press, 1966.  Google Scholar

[19]

Discrete Contin. Dyn. Syst., 9 (2003), 383-397.  Google Scholar

[20]

Integral Equations Operator Theory, 32 (1998), 332-353. doi: 10.1007/BF01203774.  Google Scholar

[21]

J. Math. Anal. Appl., 261 (2001), 28-44. doi: 10.1006/jmaa.2001.7450.  Google Scholar

[22]

Proc. Amer. Math. Soc., 137 (2009), 3025-3035. doi: 10.1090/S0002-9939-09-09871-2.  Google Scholar

[23]

Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.  Google Scholar

[24]

Appl. Math. Lett., 17 (2004), 779-783. doi: 10.1016/j.aml.2004.06.005.  Google Scholar

[25]

Lecture Notes in Mathematics, 2002, Springer, 2010. doi: 10.1007/978-3-642-14258-1.  Google Scholar

[26]

J. Difference Equ. Appl., 10 (2004), 1085-1105. doi: 10.1080/10236190412331314178.  Google Scholar

[27]

Dynam. Contin. Discrete Impuls. Systems, 13 (2006), 1-26.  Google Scholar

[28]

J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047.  Google Scholar

[29]

J. Math. Anal. Appl., 323 (2006), 1465-1478. doi: 10.1016/j.jmaa.2005.12.002.  Google Scholar

[30]

J. Math. Anal. Appl., 344 (2008), 906-920. doi: 10.1016/j.jmaa.2008.03.019.  Google Scholar

[31]

Adv. Difference Equ., (2010), Article ID 869608, 24 pp.  Google Scholar

[32]

Integral Equations Operator Theory, 66 (2010), 113-140 doi: 10.1007/s00020-009-1735-5.  Google Scholar

[33]

J. Math. Anal. Appl., 380 (2011), 17-32. doi: 10.1016/j.jmaa.2011.02.045.  Google Scholar

show all references

References:
[1]

J. Difference Equ. Appl., 15 (2009), 659-678. doi: 10.1080/10236190802259824.  Google Scholar

[2]

J. Difference Equ. Appl., 2 (1996), 251-262. doi: 10.1080/10236199608808060.  Google Scholar

[3]

Nonlinear Analysis, 72 (2010), 881-893. doi: 10.1016/j.na.2009.07.028.  Google Scholar

[4]

Discrete Contin. Dyn. Syst., 30 (2011), 39-53. doi: 10.3934/dcds.2011.30.39.  Google Scholar

[5]

Discrete Contin. Dyn. Syst., 32 (2012), 1537-1555. doi: 10.3934/dcds.2012.32.1537.  Google Scholar

[6]

Bull. Sci. Math., 136 (2012), 277-290. doi: 10.1016/j.bulsci.2011.12.003.  Google Scholar

[7]

Pure Appl. Math., 129, 1988.  Google Scholar

[8]

J. Math. Anal. Appl., 304 (2005), 511-530. doi: 10.1016/j.jmaa.2004.09.042.  Google Scholar

[9]

J. Difference Equ. Appl., 17 (2011), 657-675. doi: 10.1080/10236190903146938.  Google Scholar

[10]

J. Difference Equ. Appl., 18 (2012), 909-939. doi: 10.1080/10236198.2010.531276.  Google Scholar

[11]

J. Differential Equations, 120 (1995), 429-477. doi: 10.1006/jdeq.1995.1117.  Google Scholar

[12]

Proc. Amer. Math. Soc., 124 (1996), 1071-1081. doi: 10.1090/S0002-9939-96-03433-8.  Google Scholar

[13]

Math. Ann., 172 (1967), 139-166.  Google Scholar

[14]

Springer Verlag, Berlin, Heidelberg, New-York, 1978.  Google Scholar

[15]

J. Difference Equ. Appl., 3 (1998), 417-448. doi: 10.1080/10236199708808113.  Google Scholar

[16]

Adv. Appl. Math., 31 (2003), 1-9. doi: 10.1016/S0196-8858(03)00072-1.  Google Scholar

[17]

Comput. Math. Appl., 42 (2001), 301-311. doi: 10.1016/S0898-1221(01)00155-9.  Google Scholar

[18]

Academic Press, 1966.  Google Scholar

[19]

Discrete Contin. Dyn. Syst., 9 (2003), 383-397.  Google Scholar

[20]

Integral Equations Operator Theory, 32 (1998), 332-353. doi: 10.1007/BF01203774.  Google Scholar

[21]

J. Math. Anal. Appl., 261 (2001), 28-44. doi: 10.1006/jmaa.2001.7450.  Google Scholar

[22]

Proc. Amer. Math. Soc., 137 (2009), 3025-3035. doi: 10.1090/S0002-9939-09-09871-2.  Google Scholar

[23]

Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.  Google Scholar

[24]

Appl. Math. Lett., 17 (2004), 779-783. doi: 10.1016/j.aml.2004.06.005.  Google Scholar

[25]

Lecture Notes in Mathematics, 2002, Springer, 2010. doi: 10.1007/978-3-642-14258-1.  Google Scholar

[26]

J. Difference Equ. Appl., 10 (2004), 1085-1105. doi: 10.1080/10236190412331314178.  Google Scholar

[27]

Dynam. Contin. Discrete Impuls. Systems, 13 (2006), 1-26.  Google Scholar

[28]

J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047.  Google Scholar

[29]

J. Math. Anal. Appl., 323 (2006), 1465-1478. doi: 10.1016/j.jmaa.2005.12.002.  Google Scholar

[30]

J. Math. Anal. Appl., 344 (2008), 906-920. doi: 10.1016/j.jmaa.2008.03.019.  Google Scholar

[31]

Adv. Difference Equ., (2010), Article ID 869608, 24 pp.  Google Scholar

[32]

Integral Equations Operator Theory, 66 (2010), 113-140 doi: 10.1007/s00020-009-1735-5.  Google Scholar

[33]

J. Math. Anal. Appl., 380 (2011), 17-32. doi: 10.1016/j.jmaa.2011.02.045.  Google Scholar

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