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Nonuniform h-dichotomy with strong invariant projections for discrete time systems in Banach spaces
Corresponding Author(s) : Carmen-Florinela Popa
Student Thinkers and Advanced Research,
Vol. 3 No. 1 (2024): Proceedings of the 7th International Conference XGEN
Abstract
The present paper treats two concepts of h-dichotomy in the nonuniform case for discrete-time systems in Banach spaces, that is h-dichotomy and weak h-dichotomy. These concepts use strongly invariant types of h-dichotomy projections sequences. The main result of the paper is characterizations of Datko-types for these concepts.
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- R.P. Agarwal, "Difference Equations and Inequalities", Marcel Dekker, New York, 1992
- M.G. Babuµia, M. I. Kovacs, M. L p dat, M. Megan, Discrete (h;k)-dichotomy and remarks on the boundedness of the projections, J. Oper. Article ID 196345, 2014, p. 6.
- M.G. Babuµia, M. Megan, I.-L. Popa, On (h,k)-dichotomies for nonautonomous difference equations in Banach spaces, Int. J. Differ. Equ., 2013, 7, (761680).
- M.G. Babuµia, M. Megan, Nonuniform exponential dichotomy for noninvertible evolution operators in Banach Spaces, An. Stiint. Univ. Al. I. Cuza Iasi Mat. (N.S.) Tomul LXII, f. 2, (2016), vol. 3, 771-799.
- M.G. Babuµia, M. Megan, Nonuniform exponential dichotomy for discrete dynamical systems in Banach Spaces, Mediterr. J. Math, 2016, 13, 1653-1667.
- L. Barreira, C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity, J. Dynam. Differential Equations, 2009, 19, 215-241
- L. Barreira, C. Valls, Nonuniform exponential dichotomies and admissibility, Discrete Contin. Dyn. Syst., 2011, 30(1), 39-53.
- R. Boruga(Toma), M. Megan, Datko type characterizations for nonuniform polynomial dichotomy, Carpathian J. Math, 2021, 37(1), 45-51.
- R. Borug (Toma), "On Uniform Dichotomies for the Growth Rates of Linear Discrete-time Dynamical Systems in Banach Spaces". In: Olaru, S., Cushing, J., Elaydi, S., Lozi, R. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2022. Springer Proceedings in Mathematics and Statistics, vol. 444, Springer, Cham. doi: https : //doi.org/10.1007/978 − 3 − 031 − 51049 − 69
- C.V. Coffman, J.J. Sch¨affer, "Dichotomies for Linear Difference Equations", Math. Ann., 1967, 172, 139-166.
- V. Crai, M. Aldescu, On (h,k)-dichotomy of linear discrete-time systems in Banach spaces, In: Elaydi, S., Pötzsche, C., Sasu, A. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2017. Springer Proceedings in Mathematics and Statistics, vol 287.
- J. L. Daleckii, M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI, 1974
- D. Dragicevi ˇ c´, A. L. Sasu, B. Sasu, Admissibility and nonuniform polynomial dichotomies, Math Nachr, 2019, 293(2), 226-243.
- D. Dragicevi ˇ c´, A. L. Sasu, B. Sasu, On the asymptotic behavior of discrete dynamical systems - An ergodic theory approach, J. Differential Equations, 2020, 268, 4786-4829.
- D. Dragicevi ˇ c´, A. L. Sasu, B. Sasu, Admissibility and polynomial dichotomy of discrete nonautonomous systems, Carpathian Journal of Mathematics, 2020, 38, 737-762.
- S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.
- A. G in , M. Megan , C.F. Popa, Uniform Dichotomy Concepts for Discrete-Time Skew Evolution Cocycles in Banach Spaces, Mathematics, 2021, 9(17):2177 https://doi.org/10.3390/math9172177.
- Y. Latushkin, T. Radolph, R. Schnaubelt, "Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces", J. Dynam. Differential Equations, 1998, 10, 489-509.
- T. Li, "Die Stabilitatsfrage bei Differenzengleichungen", Acta Math., 1934, 63, 99141.
- J.L. Massera, J.J. Schaffer, "Linear Differential Equations and Function Spaces", Academic Press, New York, 1966.
- M. Megan, A.L. Sasu, B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete and Continuous Dynamical Systems, 2003, 9 (2), 383-398.
- M. Pinto, "Discrete Dichotomies", Computers Math. Appl., 1994, 28, 259-270.
- I-L Popa, M. Megan, T. Ceaușu, Exponential dichotomies for linear discrete-time systems in Banach spaces, Appl. Anal. Discrete Math., 2012, 6, 140-155.
- P. Preda, M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces, Bull. Austral. Math. Soc., 1983, 27, 31-52, 1983.
- A. L. Sasu, M-G. Babuµia, B. Sasu, Admissibility and nonuniform exponential dichotomy on the half-line, Bulletin des Sciences Mathematiques, 2013, 137.4: 466-484.
- A. L. Sasu, B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces, Integral Equations and Operator Theory, 2006, 54(1), 113-130.
- A.L. Sasu, B. Sasu, Admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line, Applied Mathematics and Computation, 2020, 378: 125-167.
- B. Sasu, A.L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 2013, 33, 3057-3084.
- B. Sasu, A. L. Sasu, Sisteme dinamice discrete, Editura Politehnica (ediµia 1 - 2006), 2013
- T. Yue, Nonuniform polynomial dichotomy for noninvertible linear discrete-time systems in Banach spaces, J. Control Sci. Eng., 2015.
References
R.P. Agarwal, "Difference Equations and Inequalities", Marcel Dekker, New York, 1992
M.G. Babuµia, M. I. Kovacs, M. L p dat, M. Megan, Discrete (h;k)-dichotomy and remarks on the boundedness of the projections, J. Oper. Article ID 196345, 2014, p. 6.
M.G. Babuµia, M. Megan, I.-L. Popa, On (h,k)-dichotomies for nonautonomous difference equations in Banach spaces, Int. J. Differ. Equ., 2013, 7, (761680).
M.G. Babuµia, M. Megan, Nonuniform exponential dichotomy for noninvertible evolution operators in Banach Spaces, An. Stiint. Univ. Al. I. Cuza Iasi Mat. (N.S.) Tomul LXII, f. 2, (2016), vol. 3, 771-799.
M.G. Babuµia, M. Megan, Nonuniform exponential dichotomy for discrete dynamical systems in Banach Spaces, Mediterr. J. Math, 2016, 13, 1653-1667.
L. Barreira, C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity, J. Dynam. Differential Equations, 2009, 19, 215-241
L. Barreira, C. Valls, Nonuniform exponential dichotomies and admissibility, Discrete Contin. Dyn. Syst., 2011, 30(1), 39-53.
R. Boruga(Toma), M. Megan, Datko type characterizations for nonuniform polynomial dichotomy, Carpathian J. Math, 2021, 37(1), 45-51.
R. Borug (Toma), "On Uniform Dichotomies for the Growth Rates of Linear Discrete-time Dynamical Systems in Banach Spaces". In: Olaru, S., Cushing, J., Elaydi, S., Lozi, R. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2022. Springer Proceedings in Mathematics and Statistics, vol. 444, Springer, Cham. doi: https : //doi.org/10.1007/978 − 3 − 031 − 51049 − 69
C.V. Coffman, J.J. Sch¨affer, "Dichotomies for Linear Difference Equations", Math. Ann., 1967, 172, 139-166.
V. Crai, M. Aldescu, On (h,k)-dichotomy of linear discrete-time systems in Banach spaces, In: Elaydi, S., Pötzsche, C., Sasu, A. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2017. Springer Proceedings in Mathematics and Statistics, vol 287.
J. L. Daleckii, M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI, 1974
D. Dragicevi ˇ c´, A. L. Sasu, B. Sasu, Admissibility and nonuniform polynomial dichotomies, Math Nachr, 2019, 293(2), 226-243.
D. Dragicevi ˇ c´, A. L. Sasu, B. Sasu, On the asymptotic behavior of discrete dynamical systems - An ergodic theory approach, J. Differential Equations, 2020, 268, 4786-4829.
D. Dragicevi ˇ c´, A. L. Sasu, B. Sasu, Admissibility and polynomial dichotomy of discrete nonautonomous systems, Carpathian Journal of Mathematics, 2020, 38, 737-762.
S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.
A. G in , M. Megan , C.F. Popa, Uniform Dichotomy Concepts for Discrete-Time Skew Evolution Cocycles in Banach Spaces, Mathematics, 2021, 9(17):2177 https://doi.org/10.3390/math9172177.
Y. Latushkin, T. Radolph, R. Schnaubelt, "Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces", J. Dynam. Differential Equations, 1998, 10, 489-509.
T. Li, "Die Stabilitatsfrage bei Differenzengleichungen", Acta Math., 1934, 63, 99141.
J.L. Massera, J.J. Schaffer, "Linear Differential Equations and Function Spaces", Academic Press, New York, 1966.
M. Megan, A.L. Sasu, B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete and Continuous Dynamical Systems, 2003, 9 (2), 383-398.
M. Pinto, "Discrete Dichotomies", Computers Math. Appl., 1994, 28, 259-270.
I-L Popa, M. Megan, T. Ceaușu, Exponential dichotomies for linear discrete-time systems in Banach spaces, Appl. Anal. Discrete Math., 2012, 6, 140-155.
P. Preda, M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces, Bull. Austral. Math. Soc., 1983, 27, 31-52, 1983.
A. L. Sasu, M-G. Babuµia, B. Sasu, Admissibility and nonuniform exponential dichotomy on the half-line, Bulletin des Sciences Mathematiques, 2013, 137.4: 466-484.
A. L. Sasu, B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces, Integral Equations and Operator Theory, 2006, 54(1), 113-130.
A.L. Sasu, B. Sasu, Admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line, Applied Mathematics and Computation, 2020, 378: 125-167.
B. Sasu, A.L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 2013, 33, 3057-3084.
B. Sasu, A. L. Sasu, Sisteme dinamice discrete, Editura Politehnica (ediµia 1 - 2006), 2013
T. Yue, Nonuniform polynomial dichotomy for noninvertible linear discrete-time systems in Banach spaces, J. Control Sci. Eng., 2015.