On uniform h-dichotomy of skew-evolution cocycles in Banach spaces

Găină, Ariana

doi:10.2478/awutm-2022-0020

Abstract

The paper presents some characterizations for the study of dichotomy of dynamical systems. Integral characterizations of Datko type are considered for the notion of uniform dichotomy with growth rates, also called uniform h-dichotomy, for skew-evolution cocycles in Banach spaces.

References

[1] L. Barreira, C. Valls, Polynomial growth rates, Nonlinear Anal. 71 (2009), 5208–5219.10.1016/j.na.2009.04.005
Search in Google Scholar Back to article
[2] A. Bento, N. Lupa, M. Megan, C. Silva, Integral conditions for nonuniform μ−dichotomy on the half-line, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 3063–3077.10.3934/dcdsb.2017163
Search in Google Scholar Back to article
[3] R. Boruga, M. Megan, On uniform polynomial dichotomy in Banach spaces, Bul. Ştiinţ. Univ. Politeh. Timiş., Ser. Mat.-Fiz. 63 (77), Issue 1 (2018), 32–40.
Search in Google Scholar Back to article
[4] R. Boruga (Toma), M. Megan, D. M-M. Toth, Integral characterizations for uniform stability with growth rates in Banach spaces, Axioms 10, 235 (2021), 1–12.10.3390/axioms10030235
Search in Google Scholar Back to article
[5] R. Boruga (Toma), M. Megan, Datko type characterizations for nonuniform polynomial dichotomy, Carpathian J. Math. 37 (2021), 45–51.10.37193/CJM.2021.01.05
Search in Google Scholar Back to article
[6] R. Boruga (Toma), M. Megan, D. M-M. Toth, On uniform instability with growth rates in Banach spaces, Carpathian J. Math. 38 (3) (2022), 789–796.10.37193/CJM.2022.03.22
Search in Google Scholar Back to article
[7] R. Boruga, M. Megan, On some characterizations for uniform dichotomy of evolution operators in Banach spaces, Mathematics 10 (19), 3704 (2022), 1–21.10.3390/math10193704
Search in Google Scholar Back to article
[8] W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math. Springer-Verlag, Berlin-New York, 1978.10.1007/BFb0067780
Search in Google Scholar Back to article
[9] J. L. Daleckii, M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, Trans. Math. Monographs, vol. 43 ( 1974), Amer. Math. Soc., Providence, R. I.
Search in Google Scholar Back to article
[10] R. Datko, Uniform asymptotic stability of evolutionary processes in Banach space, SIAM J. Math. Anal. 3 (1972), 428–445.10.1137/0503042
Search in Google Scholar Back to article
[11] R. Datko, The uniform asymptotic stability of certain neutral differential-difference equations, J. Math. Anal. Appl. 58 (1977), 510–526.10.1016/0022-247X(77)90189-5
Search in Google Scholar Back to article
[12] D. Dragičević, A. L. Sasu, B. Sasu, On the asymptotic behavior of discrete dynamical systems - An ergodic theory approach, J. Differential Equations 268 (2020), 4786–4829.10.1016/j.jde.2019.10.037
Search in Google Scholar Back to article
[13] D. Dragičević, A. L. Sasu, B. Sasu, On polynomial dichotomies of discrete nonautonomous systems on the half-line, Carpathian J. Math. 38 (2022), 663–680.10.37193/CJM.2022.03.12
Search in Google Scholar Back to article
[14] D. Dragičević, A. L. Sasu, B. Sasu, Admissibility and polynomial dichotomy of discrete nonautonomous systems, Carpathian J. Math. 38 (2022), 737–762.10.37193/CJM.2022.03.18
Search in Google Scholar Back to article
[15] A. Găină, M. Megan, C. F. Popa, Uniform dichotomy concepts for discrete-time skew evolution cocycles in Banach spaces, Mathematics 9 (17), 2177 (2021), 1–11.10.3390/math9172177
Search in Google Scholar Back to article
[16] P. V. Hai, Polynomial stability and polynomial instability for skew-evolution semiflows, Results Math. 74, 175 (2019), 1–19.10.1007/s00025-019-1099-3
Search in Google Scholar Back to article
[17] W. Littman, A generalization of a theorem of Datko and Pazy, Advances in Computing and Control, Lecture Notes in Control and Inform. Sci. 130, Springer-Verlag, Berlin, 1989, 318–323.10.1007/BFb0043280
Search in Google Scholar Back to article
[18] N. Lupa, M. Megan, Exponential dichotomies of evolution operators in Banach spaces, Monatsh. Math. 174 (2014), 265–284.10.1007/s00605-013-0517-y
Search in Google Scholar Back to article
[19] J. L. Massera, J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure Appl. Math. 21 Academic Press, New York-London, 1966.
Search in Google Scholar Back to article
[20] M. Megan, On H- stability of evolution operators, Qualitative Problems for Differential Equations and Control Theory, World Sci. Publishing, 1995, 33–40.
Search in Google Scholar Back to article
[21] M. Megan, On (h, k)-dichotomy of evolution operators in Banach spaces, Dyn. Syst. Appl. 5 (2) (1996), 189–195.
Search in Google Scholar Back to article
[22] M. Megan, A. L. Sasu, B. Sasu, Stabilizability and controlability of systems associated to linear skew-product semiflows, Rev. Mat. Complut. 15 (2) (2002), 599–618.10.5209/rev_REMA.2002.v15.n2.16932
Search in Google Scholar Back to article
[23] M. Megan, A. L. Sasu, B. Sasu, Uniform exponential dichotomy and admissibility for linear skew-product semiflows, Oper. Theory Adv. Appl. 153 (2004), 185–195.10.1007/3-7643-7314-8_11
Search in Google Scholar Back to article
[24] M. Megan, A. L. Sasu, B. Sasu, Exponential stability and exponential instability for linear skew-product flows, Math. Bohem. 129 (3) (2004), 225–243.10.21136/MB.2004.134146
Search in Google Scholar Back to article
[25] M. Megan, C. Stoica, L. Buliga, On asymptotic behaviors for linear skew evolution semi-flows in Banach spaces, Carpathian J. Math. 23 (2007), 117–125.
Search in Google Scholar Back to article
[26] M. Megan, C. Stoica, Discrete asymptotic behaviors for skew-evolution semiflows on Banach spaces, Carpathian J. Math. 24 (2008), 348–355.
Search in Google Scholar Back to article
[27] M. Megan, C. Stoica, Concepts of dichotomy for skew-evolution semiflows in Banach spaces, Ann. Acad. Rom. Sci. Ser. Math. Appl. 2 (2) (2010), 125–140.
Search in Google Scholar Back to article
[28] C. L. Mihiţ, M. Lăpădat, On uniform polynomial dichotomy of skew-evolution semiflows on the half-line, Bul.Ştiinţ. Univ. Politeh. Timiş., Ser. Mat.-Fiz. 62 (2017), 54–61.
Search in Google Scholar Back to article
[29] J. van Neerven, Exponential stability of operators and operator semigroups, J. Funct. Anal. 130 (1995), 293–309.10.1006/jfan.1995.1071
Search in Google Scholar Back to article
[30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1983.10.1007/978-1-4612-5561-1
Search in Google Scholar Back to article
[31] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703–728.10.1007/BF01194662
Search in Google Scholar Back to article
[32] A. P. Petre, M. Megan, On uniform exponential dichotomy of linear skew-product three-parameter semiflows in Banach spaces, ROMAI J. 7 (1) (2011), 141–150.
Search in Google Scholar Back to article
[33] M. Pinto, Asymptotic integration of a system resulting from the perturbation of an h-system, J. Math. Anal. Appl. 131 (1988), 194–216.10.1016/0022-247X(88)90200-4
Search in Google Scholar Back to article
[34] S. Rolewicz, On uniform N - equistability, J. Math. Anal. Appl. 115 (1986), 434–441.10.1016/0022-247X(86)90006-5
Search in Google Scholar Back to article
[35] R. J. Sacker, G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems II, J. Differential Equations 22 (1976), 478–496.10.1016/0022-0396(76)90042-5
Search in Google Scholar Back to article
[36] B. Sasu, Integral conditions for exponential dichotomy: A nonlinear approach, Bull. Sci. Math. 134 (2010), 235–246.10.1016/j.bulsci.2009.06.006
Search in Google Scholar Back to article
[37] A. L. Sasu, B. Sasu, Integral equations, dichotomy of evolution families on the half-line and applications, Integral Equations Operator Theory 66 (1) (2010), 113–140.10.1007/s00020-009-1735-5
Search in Google Scholar Back to article
[38] A. L. Sasu, M. Megan, B. Sasu, On Rolewicz-Zabczyk techniques in the stability theory of dynamical systems, Fixed Point Theory 13 (2012), 205–236.
Search in Google Scholar Back to article
[39] C. Stoica, M. Megan, On skew-evolution semiflows in infinite dimensional spaces, SIAM International Conference on Emerging Topics in Dynamical Systems and Partial Differential Equations DSPDEs10, Barcelona, Spain, June 2010.
Search in Google Scholar Back to article
[40] L. Zhou, K. Lu, W. Zhang, Roughness of tempered exponential dichotomies for infinite-dimensional random difference equations, J. Differential Equations 254 (2013), 4024–4046.10.1016/j.jde.2013.02.007
Search in Google Scholar Back to article
[41] L. Zhou, K. Lu, W. Zhang, Equivalences between nonuniform exponential dichotomy and admissibility, J. Differential Equations 262 (2017), 682–747.10.1016/j.jde.2016.09.035
Search in Google Scholar Back to article

On uniform h-dichotomy of skew-evolution cocycles in Banach spaces

Abstract

Paradigm

My account