Journal of East China Normal University(Natural Science) >

2020 , Vol. 2020 >Issue 6: 30 - 37

DOI: https://doi.org/10.3969/j.issn.1000-5641.201911042

Mathematics

Datko-Pazy theorem for nonuniform exponential expansiveness of linear skew-product semiflows

  • YUE Tian ,
  • SONG Xiaoqiu
Expand
  • 1. School of Sciences, Hubei University of Automotive Technology, Shiyan Hubei 442002, China;
    2. School of Mathematics, China University of Mining and Technology, Xuzhou Jiangsu 221116, China

Received date: 2019-10-12

  Online published: 2020-12-01

Fold

Abstract

In this paper, the nonuniform exponential expansiveness of linear skew-product semiflows is studied in Banach spaces based on Lyapunov norms. Some continuous and discrete versions of necessary and sufficient conditions for nonuniform exponential expansiveness are obtained via Datko-Pazy methods. The obtained conclusions are generalizations of well-known results in exponential stability and exponential dichotomy theory (Datko, Pazy, Preda et al.). Herein, we apply the main results to the study of nonuniform exponential dichotomy of linear skew-product semiflows.

Key words： linear skew-product semiflows; nonuniform exponential expansiveness; nonuniform exponential dichotomy; Datko-Pazy theorem

Cite this article

YUE Tian , SONG Xiaoqiu . Datko-Pazy theorem for nonuniform exponential expansiveness of linear skew-product semiflows[J]. Journal of East China Normal University(Natural Science), 2020 , 2020(6) : 30 -37 . DOI: 10.3969/j.issn.1000-5641.201911042

References

[1] DATKO R. Extending a theorem of A. M. Liapunov to Hilbert spaces [J]. J Math Anal Appl, 1970, 32(3): 610-616
[2] DATKO R. Uniform asymptotic stability of evolutionary processes in Banach space [J]. SIAM J Math Anal, 1972, 3(3): 428-445
[3] PAZY A. On the applicability of Lyapunov’s theorem in Hilbert space [J]. SIAM J Math Anal, 1972, 3(2): 291-294
[4] 岳田, 雷国梁, 宋晓秋. 线性斜演化半流一致指数膨胀性的若干刻画 [J]. 数学进展, 2016, 45(3): 433-442
[5] 岳田, 宋晓秋. Banach空间中GC(0, e) 类广义发展算子的一致指数不稳定性 [J]. 中山大学学报(自然科学版), 2018, 57(5): 150-154
[6] MUREŞAN M, PREDA C, PREDA P. Individual stability and instability for evolutionary processes [J]. Acta Math Hungar, 2017, 153(1): 16-23
[7] BARREIRA L, VALLS C. Admissibility for nonuniform exponential contractions [J]. J Differ Equ, 2010, 249(11): 2889-2904
[8] PREDA P, PREDA C, MORARIU C. Exponential stability concepts for evolution families on \scriptsize $ {\mathbb{R}} $ \normalsize [J]. Monatsh Math, 2015, 178(4): 611-631
[9] 岳田, 宋晓秋. 线性斜积半流的一致指数稳定性的若干刻画 [J]. 浙江大学学报(理学版), 2018, 45(5): 545-548
[10] BǍTǍRAN F, PREDA C, PREDA P. Extending some results of L. Barreira and C. Valls to the case of linear skew-product semiflows [J]. Results Math, 2017, 72(1): 965-978
[11] PREDA C, PREDA P, BǍTǍRAN F. An extension of a theorem of R. Datko to the case of (non)uniform exponential stability of linear skew-product semiflows [J]. J Math Anal Appl, 2015, 425(2): 1148-1154
[12] PREDA C, ONOFREI O R. Nonuniform exponential dichotomy for linear skew-product semiflows over semiflows [J]. Semigroup Forum, 2018, 96(2): 241-252
Options
Outlines

/

〈 〉