Skip to main content

An inequality concerning the growth bound of an evolution family and the norm of a convolution operator

Abstract

Let \(\mathcal {U}=\{U(t,s)\}_{t\ge s\ge 0}\) be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let \(\mathcal {X}\) be a certain Banach function space of X-valued functions. We prove that the growth bound of the family \(\mathcal {U}\) is less than or equal to \(-\frac{1}{c(\mathcal {U}, \mathcal {X})}\) provided that the convolution operator \(f\mapsto \mathcal {U}*f\) acts on \(\mathcal {X}.\) It is well known that under the latter assumption, the convolution operator is bounded and then \(c(\mathcal {U}, \mathcal {X})\) denotes (ad-hoc) its norm in \(\mathcal {L}(\mathcal {X}).\) As a consequence, we prove that if \(\sup \nolimits _{s\ge 0}\int \nolimits _{s}^\infty \Vert U(t,s)\Vert dt=u_1(\mathcal {U})<\infty ,\) then \(\omega _0(\mathcal {U})u_1(\mathcal {U})\le -1.\) Finally, we give an example showing that the accuracy of the estimates may be quite accurate.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Arendt, W., Batty, Ch J.K., Hieber si, M., Neubrabder, F.: Vector-valved Laplace transforms and cauchy problems, 2nd edn. Birkhuser, Basel (2011)

    Book  Google Scholar 

  2. 2.

    Buşe, C.: On the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces. NZ J Math Auckland 27, 183–190 (1998)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Buşe, C.: The spectral mapping theorem for evolution semigroups on the space of asymptotically almost periodic functions defined on the half line. Elect J Diff Eq 2002(70), 1–11 (2002)

    MATH  Google Scholar 

  4. 4.

    Buşe, C., Khan, A., Rahmat, G., Tabassum, A.: A new estimation of the growth bound of a periodic evolution family on Banach spaces, J Funct Spaces Appl. 2013(260290), 6 doi:10.1155/2013/260920

  5. 5.

    Buşe, C., Khan, A., Rahmat, G.: Uniform exponential stability for discrete non-autonomous systems via discrete evolution semigroups. Bull. Math. Soc. Sci. Math. Roumanie, Tome 57(105)(2), 193–205 (2014)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Buşe, C., O’Regan, D., Saierli, O.:An inequality concerning the growth bound of a discrete evolution family on a complex Banach space. J. Diff. Eqs. Appl. Published online: (21 Mar 2016), doi:10.1080/10236198.2016.1162160

  7. 7.

    Chicone, C., Latushkin, Y.: Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, Providence R. I. (1999)

    Book  MATH  Google Scholar 

  8. 8.

    Clark, S.: Yuri Latushkin, S. Montgomery-Smith, Timothy Randolph, Stability Radius and Internal Versus External Stability in Banach Spaces: An evolution Semigroup Approach, SIAM Journal of Control and Optimization 38(6), 1757–1793 (2000)

    Google Scholar 

  9. 9.

    Corduneanu, C.: Almost Periodic Oscillations and Waves. Springer Sciences+Business Media LLC, (2009)

  10. 10.

    Datko, R.: Uniform Asymptotic Stability of Evolutionary proccesses in a Banach Space. SIAM J. Math. Anal. 3, 428–445 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Engel, K., Nagel, R.: One-parameter semigroups for linear evolution equations. Springer-Verlag, New-York (2000)

    MATH  Google Scholar 

  12. 12.

    Gil’, M.: Integrally small perturbations of semigroups and stability of partial differential equations. Int. J. Partial Diff. Eqs. 2013(207581) doi:10.1155/2013/207581

  13. 13.

    Goldstein, J.: Semigroups of linear operators and applications. Oxford University Press, New York (1985)

    MATH  Google Scholar 

  14. 14.

    Helffer, B., Sjöstrand, J.: From resolvent bounds to semigroup bounds, preprint (2010), arXiv:1001.4171v1

  15. 15.

    Yuri, L., Valerian, Y.: Stability estimates for semigroups on Banach spaces. Discrete Contin. Dyn. Syst. 33(11–12), 5203–5216 (2013)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Levitan, B.M., Zhicov, V.V.: Almost periodic functions and differential equations. Cambridge Univ. Press, Cambridge (1982)

    Google Scholar 

  17. 17.

    Van Minh, N.: F. Räbiger, R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line. Integral equations operator theory 32, 332–353 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    van Neerven, J.: Characterization of exponentialstability of a semigroup of operators in terms of its action by convolution on vector-valued function space over \(\mathbb{R}_+\). J. Diff. Eqs. 124(2), 324–342 (1996)

    Article  MATH  Google Scholar 

  19. 19.

    van Neerven, J.M.A.M.: The asymptotic behavior of semigroups of bounded linear operators, operator theory, Adv. Appl. 88. Birkhäuser Verlag, (1996)

  20. 20.

    Rau, R.T.: Hyperbolic evolution semigroups on vector valued function spaces. Semigroup Forum 48(1), 107–118 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Räbiger, F., Rhandi, A., Schnaubelt, R.: Perturbation and an abstract characterization of evolution semigroups. J. Math. Anal. Appl. 198, 516–533 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Räbiger, F., Schnaubelt, R.: The spectral mapping theorem for evolution semigroups on spaces of vector valued functions. Semigroup Forum 52(1), 225–239 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Schnaubelt, R.: Well-posedness and asymptotic behavior of non-autonomous linear evolution equations, Evolution equations, semigroups and functional analysis, Progr. Nonlinear Diff. Eqs. Appl. 50, Birkhäuser, Basel, (2002), pp. 311–338

  24. 24.

    Schnaubelt, R.: Exponential bounds and hyperbolicity for evolution families, PhD Thesis, Tübingen (1996)

  25. 25.

    Weiss, G.: Weak \(L^p\)-stability of linear semigroup on a Hilbert space implies exponential stability. J. Diff. Equations 76, 269–285 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Zabczyk, J.: Mathematical control theory: an introduction. Birkhäuser, systems and control (1992)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Olivia Saierli.

Additional information

Communicated by Jerome A. Goldstein.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Buşe, C., O’Regan, D. & Saierli, O. An inequality concerning the growth bound of an evolution family and the norm of a convolution operator. Semigroup Forum 94, 618–631 (2017). https://doi.org/10.1007/s00233-016-9822-9

Download citation

Keywords

  • Uniform exponential stability
  • Growth bounds
  • Exponentially bounded evolution families of operators
  • Convolution operator on function spaces
  • One dimensional heat equation