Skip to main content
SpringerLink
Log in

Almost automorphic solutions to integral equations on the line

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

Given a∈L 1(ℝ) and A the generator of an L 1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫ t−∞ a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L 1(ℝ) positive, nonincreasing and log-convex is already sufficient.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. Ser. A: Theory Methods Appl. 69(11), 3692–3705 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

  3. Basit, B.: Harmonic analysis and asymptotic behavior of solutions to the abstract Cauchy problem. Semigroup Forum 54, 58–74 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Basit, B., Pryde, A.J.: Asymptotic behavior of orbits of C 0-semigroups and solutions of linear and semilinear abstract differential equations. Russ. J. Math. Phys. 13(1), 13–30 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bazhlekova, E.: Fractional evolution equations in Banach spaces. Ph.D. Thesis, Eindhoven University of Technology (2001)

  6. Bugajewski, D., Diagana, T.: Almost automorphy of the convolution operator and applications to differential and functional differential equations. Nonlinear Stud. 13(2), 129–140 (2006)

    MATH  MathSciNet  Google Scholar 

  7. Clément, Ph., Da Prato, G.: Existence and regularity results for an integral equation with infinite delay in a Banach space. Integr. Equ. Oper. Theory 11, 480–500 (1988)

    Article  MATH  Google Scholar 

  8. Diagana, T.: Some remarks on some second-order hyperbolic differential equations. Semigroup Forum 68, 357–364 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Diagana, T., N’Guérékata, G.M.: Almost automorphic solutions to semilinear evolution equations. Funct. Differ. Equ. 13(2), 195–206 (2006)

    MATH  MathSciNet  Google Scholar 

  10. Goldstein, J.A., N’Guérékata, G.M.: Almost automorphic solutions of semilinear evolution equations. Proc. Am. Math. Soc. 133(8), 2401–2408 (2005)

    Article  MATH  Google Scholar 

  11. Gorenflo, R., Mainardi, F.: On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, 283–299 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gripenberg, G., Londen, S.-O., Staffans, O.: Volterra Integral and Functional Equations. Encyclopedia of Mathematics and Applications, vol. 34. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  13. Liang, J., Zhang, J., Xiao, T.J.: Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J. Math. Anal. Appl. 340(2), 1493–1499 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu, J., N’Guérékata, G.M., van Minh, N.: Almost automorphic solutions of second order evolution equations. Appl. Anal. 84(11), 1173–1184 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lizama, C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lizama, C.: On approximation and representation of k-regularized resolvent families. Integr. Equ. Oper. Theory 41(2), 223–229 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lizama, C., Sánchez, J.: On perturbation of k-regularized resolvent families. Taiwan. J. Math. 7(2), 217–227 (2003)

    MATH  Google Scholar 

  18. Lizama, C., Poblete, V.: On multiplicative perturbation of integral resolvent families. J. Math. Anal. Appl. 327(2), 1335–1359 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. N’Guerekata, G.M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Acad/Plenum, New York-Boston-Moscow-London (2001)

    MATH  Google Scholar 

  20. N’Guérékata, G.M.: Existence and uniqueness of almost automorphic mild solutions of some semilinear abstract differential equations. Semigroup Forum 69, 80–86 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. N’Guérékata, G.M.: Topics in Almost Automorphy. Springer, New York (2005)

    MATH  Google Scholar 

  22. Nunziato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–204 (1971)

    MATH  MathSciNet  Google Scholar 

  23. Prüss, J.: Evolutionary Integral Equations and Applications. Monographs Math., vol. 87. Birkhäuser, Basel (1993)

    MATH  Google Scholar 

  24. Shaw, S.Y., Chen, J.C.: Asymptotic behavior of (a,k)-regularized families at zero. Taiwan. J. Math. 10(2), 531–542 (2006)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Pernambuco, Av. Prof. Luiz Freire, S/N, Recife, PE, CEP. 50540-740, Brazil

    Claudio Cuevas

  2. Departamento de Matemática, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago, Chile

    Carlos Lizama

Authors
  1. Claudio Cuevas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlos Lizama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Lizama.

Additional information

Communicated by Jerome A. Goldstein.

The first author is partially supported by CNPQ/Brazil under Grant 300068/2005-0.

The second author is partially financed by Laboratorio de Análisis Estocástico, Proyecto Anillo PBCT-ACT-13.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cuevas, C., Lizama, C. Almost automorphic solutions to integral equations on the line. Semigroup Forum 79, 461–472 (2009). https://doi.org/10.1007/s00233-009-9154-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-009-9154-0

Keywords

Search

Navigation