On some growth concepts for dichotomic behaviors of evolution operators

Toma, Rovana Boruga

doi:10.2478/awutm-2022-0017

Abstract

The aim of the present paper is to emphasize some growth concepts for the dichotomic behavior of evolution operators in Banach spaces. In fact, we approach the exponential growth, the polynomial growth and the h-growthfor both uniform and nonuniform cases. Connections between concepts are established. Majorization criteria for the uniform behaviors are given.

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.J.G. Bento, C. Silva, Stable manifolds for nonuniform polynomial dichotomies, J. Funct. Anal. 257 (2009), 122–148.10.1016/j.jfa.2009.01.032
Search in Google Scholar Back to article
[3] R. Boruga (Toma), M. Megan, On some characterizations for uniform dichotomy of evolution operators in Banach spaces, Mathematics 10 (2022), 3704.10.3390/math10193704
Search in Google Scholar Back to article
[4] R. Boruga, 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
[5] R. Boruga (Toma), M. Megan, Nonuniform polynomial dichotomy with Lyapunov type norms, Ann. Acad. Rom. Sci. Ser. Math. Appl. 12 (1-2) (2020), 329–343.10.56082/annalsarscimath.2020.1-2.329
Search in Google Scholar Back to article
[6] 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
[7] R. Boruga (Toma), M. Megan, D. M.-M. Toth, Integral characterizations for uniform stability with growth rates in Banach spaces, Axioms 10 (2021), 235.10.3390/axioms10030235
Search in Google Scholar Back to article
[8] R. Boruga, M. Megan, Nonuniform polynomial dichotomy with invariant projection families, The 15th ICMA Conf. Proc., Politehnica University of Timisoara (2018), 30–46.
Search in Google Scholar Back to article
[9] R. Boruga (Toma), D. I. Borlea (Pătraşcu), D. M.-M. Toth, On uniform stability with growth rates in Banach spaces, 2021 IEEE 15th International Symposium on Applied Computational Intelligence and Informatics (SACI) (2021), pp. 000393–000396.10.1109/SACI51354.2021.9465607
Search in Google Scholar Back to article
[10] R. Boruga, M. Megan, On uniform polynomial dichotomy in Banach spaces, Bul. Ştiinţ. Univ. Politeh. Timiş., Ser. Mat.-Fiz. 63(77) (2018), 32–40.
Search in Google Scholar Back to article
[11] V. Crai, M. Megan, (h, k)-dichotomy and Lyapunov type norms, An. Univ. Vest Timiş. Ser. Mat.-Inform. 56 (2) (2018), 115–130.10.2478/awutm-2018-0019
Search in Google Scholar Back to article
[12] 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
[13] 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
[14] A. Găină, M. Megan, C.F. Popa, Uniform dichotomy concepts for discrete-time skew evolution cocycles in Banach spaces, Mathematics 9 (2021), 2177.10.3390/math9172177
Search in Google Scholar Back to article
[15] P. V. Hai, On the polynomial stability of evolution families, Appl. Anal. 95 (2016), 1239–1255.10.1080/00036811.2015.1058364
Search in Google Scholar Back to article
[16] P. V. Hai, Polynomial stability and polynomial instability for skew-evolution semiflows, Results Math. 175 (2019), 1–19.10.1007/s00025-019-1099-3
Search in Google Scholar Back to article
[17] N. Lupa, Comportări asimptotice exponenţiale pentru operatori de evoluţie n spaţii Banach, Editura Politehnica, Timişoara, 2014.
Search in Google Scholar Back to article
[18] 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
[19] M. Megan, On H- stability of evolution operators, Qualitative Problems for Differential Equations and Control Theory, World Scientific Publishing (1995), 33–40.
Search in Google Scholar Back to article
[20] 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
[21] M. Megan, C. Stoica, On uniform exponential trichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory 60 (4) (2008), 499–506.10.1007/s00020-008-1555-z
Search in Google Scholar Back to article
[22] M. Megan, B. Sasu, A.L. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations and Operator Theory 44 (2002), 71–78.10.1007/BF01197861
Search in Google Scholar Back to article
[23] M. Megan, A.L. Sasu, B. Sasu, On uniform exponential dichotomy of linear skew product semiflows, Bull. Belg. Math. Soc. Simon Stevin 10 (2003), 1–21.10.36045/bbms/1047309409
Search in Google Scholar Back to article
[24] M. Megan, A. L. Sasu, B. Sasu, The Asymptotic Behavior of Evolution Families, Ed. Mirton 2003.
Search in Google Scholar Back to article
[25] M. Megan, A. L. Sasu, B. Sasu, Nonuniform exponential unstability of evolution operators in Banach spaces, Glas. Mat. 56 (2001), 287–296.
Search in Google Scholar Back to article
[26] C. L. Mihiţ, On uniform h-stability of evolution operators in Banach spaces, Theory Appl. Math. Comput. Sci. 6 (2016), 19–27.
Search in Google Scholar Back to article
[27] M. Pinto, Perturbations of asymptotically stable differential systems, Analysis 4 (1-2) (1984), 161–176.10.1524/anly.1984.4.12.161
Search in Google Scholar Back to article
[28] M. Pinto, Asymptotic integration of a system resulting from the perturbation of an h-system, J. Math. Anal. Appl. 131 (1) (1988), 194–216.10.1016/0022-247X(88)90200-4
Search in Google Scholar Back to article
[29] M.L. Rămneanţu, T. Ceauşu, M. Megan, On nonuniform polynomial dichotomy of evolution operators in Banach spaces, Int. J. Pure Appl. Math. 75 (3) (2012), 305–318.
Search in Google Scholar Back to article
[30] M. L. Rămneanţu, M. Megan, T. Ceauşu, Polynomial instability of evolution operators in Banach spaces, Carpathian J. Math. 29 (2013), 77–83.10.37193/CJM.2013.01.06
Search in Google Scholar Back to article
[31] A. L. Sasu, M. G. Babutia, B. Sasu, Admissibility and nonuniform exponential dichotomy on the half-line, Bull. Sci. Math. 137 (2013), 466–484.10.1016/j.bulsci.2012.11.002
Search in Google Scholar Back to article

On some growth concepts for dichotomic behaviors of evolution operators

Abstract

Paradigm

My account