Skip to main content

Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay

Abstract

We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the nonlinear semigroup.

[12345678910111213141516171819202122232425]

References

  1. 1.

    Adimy M, Bouzahir H, Ezzinbi K: Local existence and stability for some partial functional differential equations with infinite delay. Nonlinear Analysis. Theory, Methods & Applications. Series A 2002,48(3):323–348. 10.1016/S0362-546X(00)00184-X

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj. Serio Internacia 1978,21(1):11–41.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Hino Y, Murakami S, Naito T: Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics. Volume 1473. Springer, Berlin, Germany; 1991:x+317.

    Google Scholar 

  4. 4.

    Brendle S, Nagel R: PFDE with nonautonomous past. Discrete and Continuous Dynamical Systems. Series A 2002,8(4):953–966.

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.

    Google Scholar 

  6. 6.

    Adimy M, Bouzahir H, Ezzinbi K: Existence for a class of partial functional differential equations with infinite delay. Nonlinear Analysis. Theory, Methods & Applications. Series A 2001,46(1):91–112. 10.1016/S0362-546X(99)00447-2

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Benkhalti R, Bouzahir H, Ezzinbi K: Existence of a periodic solution for some partial functional-differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2001,256(1):257–280. 10.1006/jmaa.2000.7321

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Bouzahir H, Ezzinbi K: Global attractor for a class of partial functional differential equations with infinite delay. In Topics in Functional Differential and Difference Equations (Lisbon, 1999), Fields Inst. Commun.. Volume 29. Edited by: Faria T, Freitas P. American Mathematical Society, Providence, RI, USA; 2001:63–71.

    Google Scholar 

  9. 9.

    Ruess WM: Linearized stability for nonlinear evolution equations. Journal of Evolution Equations 2003,3(2):361–373.

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Ruess WM: Existence of solutions to partial functional evolution equations with delay. In Functional Analysis (Trier, 1994). Edited by: Dierolf S, Dineen S, Domański P. de Gruyter, Berlin, Germany; 1996:377–387.

    Google Scholar 

  11. 11.

    Ruess WM: Existence and stability of solutions to partial functional-differential equations with delay. Advances in Differential Equations 1999,4(6):843–876.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Ruess WM: Existence of solutions to partial functional-differential equations with delay. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.. Volume 178. Edited by: Kartsatos AG. Dekker, New York, NY, USA; 1996:259–288.

    Google Scholar 

  13. 13.

    Ruess WM, Summers WH: Linearized stability for abstract differential equations with delay. Journal of Mathematical Analysis and Applications 1996,198(2):310–336. 10.1006/jmaa.1996.0085

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Ruess WM, Summers WH: Operator semigroups for functional-differential equations with delay. Transactions of the American Mathematical Society 1994,341(2):695–719. 10.2307/2154579

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Ruess WM, Summers WH: Almost periodicity and stability for solutions to functional-differential equations with infinite delay. Differential and Integral Equations 1996,9(6):1225–1252.

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Kartsatos AG, Parrott ME: The weak solution of a functional-differential equation in a general Banach space. Journal of Differential Equations 1988,75(2):290–302. 10.1016/0022-0396(88)90140-4

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Thieme HR: Semiflows generated by Lipschitz perturbations of non-densely defined operators. Differential and Integral Equations 1990,3(6):1035–1066.

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Adimy M, Laklach M, Ezzinbi K: Non-linear semigroup of a class of abstract semilinear functional differential equations with a non-dense domain. Acta Mathematica Sinica (English Series) 2004,20(5):933–942. 10.1007/s10114-004-0341-3

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Laklach M: Contribution à l'étude des équations aux dérivées partielles à retard et de type neutre, Thèse de doctorat.

  20. 20.

    Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.

    Google Scholar 

  21. 21.

    Naito T, Minh NV, Shin JS: Spectrum and (almost) periodic solutions of functional differential equations. Vietnam Journal of Mathematics 2002,30(supplement):577–589.

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Naito T, Minh NV, Shin JS: The spectrum of evolution equations with infinite delay. unpublished unpublished

  23. 23.

    Travis CC, Webb GF: Existence and stability for partial functional differential equations. Transactions of the American Mathematical Society 1974, 200: 395–418.

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Yosida K: Functional Analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Volume 123. 6th edition. Springer, Berlin, Germany; 1980:xii+501.

    Google Scholar 

  25. 25.

    Crandall MG, Liggett TM: Generation of semi-groups of nonlinear transformations on general Banach spaces. American Journal of Mathematics 1971, 93: 265–298. 10.2307/2373376

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hassane Bouzahir.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Bouzahir, H. Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay. J Inequal Appl 2007, 049125 (2007). https://doi.org/10.1155/2007/49125

Download citation

Keywords

  • Differential Equation
  • Phase Space
  • Linear Part
  • Functional Differential Equation
  • Resolvent Estimate
\