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CN 34-1054/N

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Some characterizations for the exponential φ-expansiveness of linear skew-product semiflows

  • 1.

    School of Science, Hubei University of Automotive Technology, Shiyan 442002, China

  • 2.

    Hubei Key Laboratory of Automotive Power Train and Electronics Control, Shiyan 442002, China

  • 3.

    School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.05.004
  • Received Date: 04 March 2019
  • Accepted Date: 27 May 2020
  • Rev Recd Date: 27 May 2020
  • Publish Date: 31 May 2020
    Abstract

  • Abstract

    The exponential φ-expansiveness of linear skew-product semiflows in Banach space was studied. Based on the definition of uniform exponential expansiveness, a linear skew-product semiflow with exponential φ-expansiveness was presented. Some characterizations for exponential φ-expansiveness were obtained via mathematical analysis and operator theory. The results extend some well-known conclusions in the exponential stability theory.

    Abstract

    The exponential φ-expansiveness of linear skew-product semiflows in Banach space was studied. Based on the definition of uniform exponential expansiveness, a linear skew-product semiflow with exponential φ-expansiveness was presented. Some characterizations for exponential φ-expansiveness were obtained via mathematical analysis and operator theory. The results extend some well-known conclusions in the exponential stability theory.
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    • References(12)
  • [1]
    DATKO R. Extending a theorem of Liapunov to Hilbert spaces[J]. J Math Anal Appl, 1970,32(3): 610-616.
    [2]
    PAZY A. Semigroups of Linear Operators and Applications to Partial Differential Equations[M]. New York: Springer, 1983.
    [3]
    ROLEWICZ S. On uniform N-equistability[J]. J Math Anal Appl, 1986, 115(2): 434-441.
    [4]
    PREDA C. On the uniform exponential stability of linear skew-product semiflows[J]. J Funct Spaces Appl, 2006, 4(2): 145-161.
    [5]
    HAI P V. Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows[J]. Nolinear Anal, 2010, 72(12): 4390-4396.
    [6]
    PREDA C, PREDA P, BTRAN 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.
    [7]
    PREDA C, ONOFREI O R.Nonuniform exponential dichotomy for linear skew-product semiflows over semiflows[J]. Semigroup Forum, 2018, 96(2): 241-252.
    [8]
    MEGAN M, SASU A L, SASU B. Perron conditions for uniform exponential expansiveness of linear skew-product flows[J].Monatsh Math, 2003, 138(2): 145-157.
    [9]
    MEGAN M, SASU A L, SASU B. Exponential instability of linear skew-productsemiflows in terms of Banach function spaces[J]. Results Math, 2004, 45(3): 309-318.
    [10]
    MEGAN M, SASU A L, SASU B. Exponential stability and exponential instability for linear skew-product flows[J]. Math Bohem, 2004, 129(3): 225-243.
    [11]
    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.
    [12]
    PREDA P, POGAN A, PREDA C.Functionals on function and sequence spaces connected with the exponential stability of evolutionary processes[J]. Czechoslovak Math, 2006, 131(56): 425-435.
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    [1]
    DATKO R. Extending a theorem of Liapunov to Hilbert spaces[J]. J Math Anal Appl, 1970,32(3): 610-616.
    [2]
    PAZY A. Semigroups of Linear Operators and Applications to Partial Differential Equations[M]. New York: Springer, 1983.
    [3]
    ROLEWICZ S. On uniform N-equistability[J]. J Math Anal Appl, 1986, 115(2): 434-441.
    [4]
    PREDA C. On the uniform exponential stability of linear skew-product semiflows[J]. J Funct Spaces Appl, 2006, 4(2): 145-161.
    [5]
    HAI P V. Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows[J]. Nolinear Anal, 2010, 72(12): 4390-4396.
    [6]
    PREDA C, PREDA P, BTRAN 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.
    [7]
    PREDA C, ONOFREI O R.Nonuniform exponential dichotomy for linear skew-product semiflows over semiflows[J]. Semigroup Forum, 2018, 96(2): 241-252.
    [8]
    MEGAN M, SASU A L, SASU B. Perron conditions for uniform exponential expansiveness of linear skew-product flows[J].Monatsh Math, 2003, 138(2): 145-157.
    [9]
    MEGAN M, SASU A L, SASU B. Exponential instability of linear skew-productsemiflows in terms of Banach function spaces[J]. Results Math, 2004, 45(3): 309-318.
    [10]
    MEGAN M, SASU A L, SASU B. Exponential stability and exponential instability for linear skew-product flows[J]. Math Bohem, 2004, 129(3): 225-243.
    [11]
    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.
    [12]
    PREDA P, POGAN A, PREDA C.Functionals on function and sequence spaces connected with the exponential stability of evolutionary processes[J]. Czechoslovak Math, 2006, 131(56): 425-435.
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