Semigroup Forum

, Volume 97, Issue 3, pp 353–376 | Cite as

On a resolvent approach for perturbed semigroups and application to \(L^1\)-neutron transport theory

  • Hatem MegdicheEmail author
  • Mohamed Aziz Taoudi
Research Article


We give new sufficient and practical conditions in terms of the generators ensuring the stability of the critical or the essential type of a perturbed \(C_0\)-semigroup in general Banach spaces. We apply our theoretical results in order to investigate the control and in particular the time asymptotic behavior of solutions to a broad class of transport equations in \(L^1\)-spaces and higher dimension. Our results improve, complete and enrich several earlier works.


Perturbations Spectral analysis Critical type stability Essential type stability Resolvent approach Neutron transport theory 


  1. 1.
    Ben Amara, K., Megdiche, H., Taoudi, M.A.: On eventual norm continuity and compactness of the difference of two \(C_0\)-semigroups (submitted)Google Scholar
  2. 2.
    Bátkai, A., Kramar Fijavž, M., Rhandi, A.: Positive Operator Semigroups from Finite to Infinite Dimensions. Birkhauser, Springer, Basel (2016)zbMATHGoogle Scholar
  3. 3.
    Bàtkai, A., Maniar, L., Rhandi, A.: Regularity properties of perturbed Hille–Yosida operators and retarded differential equations. Semigroup Forum 64, 55–70 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brendle, S.: On the asymptotic behavior of perturbed strongly continuous semigroups. Math. Nachr. 226, 35–47 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brendle, S., Nagel, R., Poland, J.: On the spectral mapping theorem for perturbed strongly continuous semigroups. Arch. Math. 74, 365–378 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194. Springer, Berlin (1999)Google Scholar
  7. 7.
    Hille, H., Phillips, R.E.: Functional Analysis and Semigroups, vol. 31. American Mathematical Society Colloquium Publications, Providence (1957)zbMATHGoogle Scholar
  8. 8.
    Latrach, K., Megdiche, H., Taoudi, M.A.: A compactness result for perturbed semigroups and application to a transport model. J. Math. Anal. Appl. 359, 88–94 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Latrach, K., Megdiche, H.: Spectral properties and regularity of solutions to transport equations in slab geometry. Math. Models Appl. Sci. 29, 2089–2121 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Latrach, K.: Compactness results for transport equations and applications. Math. Models Methods Appl. Sci. 11(7), 1181–1202 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Li, M., Gu, X., Huang, F.L.: On unbounded perturbations of semigroups compactness and norm continuity. Semigroup Forum 65, 58–70 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mokhtar-Kharroubi, M., Sbihi, M.: Spectral mapping theorems for neutron transport, \(L^1\) -theory. Semigroup Forum 72, 249–282 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mokhtar-Kharroubi, M.: Optimal spectral theory of the linear Boltzmann equation. J. Funct. Anal. 226, 21–47 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mokhtar-Kharroubi, M.: On \(L^1\)-spectral theory of neutron transport. J. Differ. Integral Equ. 18, 1221–1242 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mokhtar-Kharroubi, M.: Mathematical Topics in Neutron Transport Theory, New Aspects. Series on Advances in Mathematics for Applied Sciences, vol. 46. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  16. 16.
    Mokhtar-Kharroubi, M.: Time asymptotic behaviour and compactness in neutron transport theory. Eur. J. Mech. B Fluid 11, 39–68 (1992)Google Scholar
  17. 17.
    Nagel, R., Poland, J.: The critical spectrum of a strongly continuous semigroup. Adv. Math. 152, 120–133 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nagel, R., Piazzera, S.: On the regularity properties of perturbed semigroups. In: International Workshop on Operator Theory (Cefalu‘, 1997), Rend. Circ. Mat. Palermo, vol. 56, pp. 99–110 (1998)Google Scholar
  19. 19.
    Rhandi, A.: Positivity and stability for a population equation with diffusion on \(L^1\). Positivity 2, 101–113 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sbihi, M.: A resolvent approach to the stability of essential and critical spectra of perturbed \(C_0\)-semigroups on Hilbert spaces with applications to transport theory. J. Evolut. Equ. 7, 35–58 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Song, D.: Some notes on spectral properties of \(C_0\) semigroups generated by linear transport operators. Transp. Theory Stat. Phys. 26, 233–242 (1997)CrossRefGoogle Scholar
  22. 22.
    Song, D., Greenberg, W.: Spectral properties of transport equations for slab geometry in \(L^1\) with reentry boundary conditions. Transp. Theory Stat. Phys. 30, 325–355 (2001)CrossRefGoogle Scholar
  23. 23.
    Vidav, I.: Existence and uniqueness of non negative eigenfunction of the Botltzmann operator. J. Math. Anal. Appl. 22, 144–155 (1968)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Voigt, J.: On the convex compactness property for the strong operator topology. Note di Mat. 12, 259–269 (1992)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Voigt, J.: A perturbation theorem for the essential spectral radius of strongly continuous semigroups. Mh. Math. 90, 153–161 (1980)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Weis, L.W.: A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory. J. Math. Anal. Appl. 129, 6–23 (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Algebra, Geometry and Spectral Theory, Department of Mathematics, Faculty of Sciences of SfaxUniversity of SfaxSfaxTunisia
  2. 2.Biomedical Department, Higher Institute of BiotechnologyUniversity of SfaxSfaxTunisia
  3. 3.National School of Applied SciencesCadi Ayyad UniversityMarrakechMorocco

Personalised recommendations