On (h; k)-Dichotomy and (h; k)-Trichotomy of Noninvertible Evolution Operators in Banach Spaces

Kovács, Monteola Ilona; Megan, Mihail; Mihiţ, Claudia Luminiţa

doi:10.2478/awutm-2014-0015

Abstract

The paper considers some concepts of (h, k)-dichotomy and (h, k)-trichotomy for noninvertible evolution operators in Ba- nach spaces. A characterization of the (h, k)-trichotomy of an evolution operator in terms of (h, k)-dichotomy for two associated evolution operators is given. As applications of this result, charac- terizations for nonuniform exponential trichotomy and nonuniform polynomial trichotomy are obtained.

References

[1] M. G. Babuţia and M. Megan, Exponential dichotomy concepts for evolution operators on the half-line, to appear in the Annals of the Academy of Romanian Scientists, Series on Mathematics and its Applications, 7, (2015)
Search in Google Scholar
[2] L. Barreira and C. Vails, Robustness of nonuniform exponential trichotomies in Banach spaces, J. Math. Anal. Appl., 351, (2009), 373-38110.1016/j.jmaa.2008.10.030
Search in Google Scholar
[3] L. Barreira, M. Fang, C. Vails, and J. Zhang. Robustness of nonuniform polyno¬mial dichotomies for difference equations, Topol. Method. Nonlinear An., 37, (2011), 357-386
Search in Google Scholar
[4] L. Barreira, J. Chu, and C. Vails, Robustness of nonuniform dichotomies with different growth rates, Sao Paolo Journal of Math. Sciences, 2, (2011), 203-23110.11606/issn.2316-9028.v5i2p203-231
Search in Google Scholar
[5] A. J. G. Bento and C. Silva, Stable manifolds for nonuniform polynomial di¬chotomy, J. Funct. Analysis, 257, (2009), 122-14810.1016/j.jfa.2009.01.032
Search in Google Scholar
[6] A. J. G. Bento, N. Lupa, M. Megan, and C. Silva, Integral conditions for nonuniform p- dichotomy, arXivGvl [math.DS], (12 May 2014)
Search in Google Scholar
[7] X. Chang, J. Zhang, and J. Qin, Robustness of nonuniform (/z, ^-dichotomies in Banach spaces, J. Math, Anal. Appl., 387, (2012), 583-594
Search in Google Scholar
[8] S. Elaydi and O. Hajek. Exponential trichotomy of differential systems, J. Math, Anal. Appl, 129, (1988), 362-37410.1016/0022-247X(88)90255-7
Search in Google Scholar
[9] J. Hong, R. Obaya, and A. Sanz, Existence of a class of ergotic solutions implies exponential trichotomy, Appl. Math. Lett., 12, (1999), 43-4510.1016/S0893-9659(99)00031-2
Search in Google Scholar
[10] M. I. Kovacs, M. G. Babuţia, and M. Megan, On (h,k)- dichotomy in Banach spaces, Bull. Ştiinţ. Univ. Politehnica Timişoara, Ser. Mat. Fiz., 59, (2014), 19-27
Search in Google Scholar
[11] M. I. Kovacs. On uniform exponential trichotomy in Banach spaces, Analele Univ. de Vest Timişoara, Seria Mat.-Inf, 52, (2014), 81-9310.2478/awutm-2014-0006
Search in Google Scholar
[12] M. I. Kovacs and R. Retezan, On (h,k)-trichotomy for evolution operators in Banach spaces, Annals of Tiberiu Popoviciu Seminar of Functional Equations, Ap¬proximation and Convexity, 12, (2014), 3-16
Search in Google Scholar
[13] N. Lupa and M. Megan, Exponential dichotomies of evolution operators in Banach spaces, Monatsh, Math., 174, (2014), 265-28410.1007/s00605-013-0517-y
Search in Google Scholar
[14] M. Megan, On (h, fc)-dichotomy of evolution operators in Banach spaces, Dynamic Systems Appl, 5, (1996), 189-196
Search in Google Scholar
[15] M. Megan, A. L. Sasu, and B. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations and Operator Theory, 44, (2002), 71-7810.1007/BF01197861
Search in Google Scholar
[16] M. Megan and M. G. Babuţia, Nonuniform exponential dichotomy for noninvert- ible evolution operators in Banach spaces, submitted
Search in Google Scholar
[17] M. Megan and C. Stoica, On uniform exponential trichotomy of evolution opera¬tors in Banach spaces, Integral Equations and Operator Theory, 60, (2008), 499-50610.1007/s00020-008-1555-z
Search in Google Scholar
[18] M. Megan, T. Ceauşu, and M. L. Ramneanţu, Polynomial stability of evolution operators in Banach spaces, Opuscula Mathematica, 31, (2011), 269-27710.7494/OpMath.2011.31.2.279
Search in Google Scholar
[19] M. L. Ramneanţu, T. Ceauşu, and M. Megan, On nonuniform polynomial dichotomy of evolution operators in Banach spaces. International Journal of Pure and Applied Mathematics, 75, (2012), 305-318
Search in Google Scholar
[20] C. Stoica and M. Megan, On (h, fc)-triehotomy for skew-evolution semiflows in Banach spaces. Stud. Univ. Babeş-Bolyai Math., 56, (2011), 147-156
Search in Google Scholar
[21] J. Zhang, X. Chang, and J. Wang, Existence and robustness of nonuniform (h, k. p, v) dichotomies for nonautonomous impulsive differential equations, J. Math. Anal. Appl, 400, (2013), 710-723 10.1016/j.jmaa.2012.12.007
Search in Google Scholar

On (h; k)-Dichotomy and (h; k)-Trichotomy of Noninvertible Evolution Operators in Banach Spaces

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