Skip to main content
SpringerLink
Log in

Bounded mild solutions to fractional integro-differential equations in Banach spaces

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

We study the existence and uniqueness of bounded solutions for the semilinear fractional differential equation

$$D^\alpha u(t)= Au(t)+ \int_{-\infty}^t a(t-s)Au(s)ds+ f \bigl(t,u(t) \bigr), \quad t \in\mathbb{R}, $$

where A is a closed linear operator defined on a Banach space X, α>0, a∈L 1(ℝ+) is a scalar-valued kernel and f:ℝ×X→X satisfies some Lipschitz type conditions. Sufficient conditions are established for the existence and uniqueness of an almost periodic, almost automorphic and asymptotically almost periodic solution, among other.

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

Fig. 1

Similar content being viewed by others

References

  1. Abbas, S.: Pseudo almost automorphic solutions of fractional order neutral differential equation. Semigroup Forum 81, 393–404 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal, R., Lakshmikantham, V., Nieto, J.: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 72, 2859–2862 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agrawal, O., Sabatier, J., Tenreiro, J.: Advances in Fractional Calculus. Springer, Dordrecht (2007)

    MATH  Google Scholar 

  4. Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69, 3692–3705 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arendt, W., Batty, C.: Almost periodic solutions of first and second-order Cauchy problems. J. Differ. Equ. 137, 363–383 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arendt, W., Batty, C.: Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line. Bull. Lond. Math. Soc. 31, 291–304 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monogr. Math., vol. 96. Birkhäuser, Basel (2001)

    Book  MATH  Google Scholar 

  8. 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, 13–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bu, S.: Well-posedness of fractional differential equations on vector-valued function spaces. Integral Equ. Oper. Theory 71, 259–274 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Chen, C., Li, M.: On fractional resolvent operator functions. Semigroup Forum 80, 121–142 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cuevas, C., Lizama, C.: Almost automorphic solutions to a class of semilinear fractional differential equations. Appl. Math. Lett. 21, 1315–1319 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Diagana, T.: Existence of solutions to some classes of partial fractional differential equations. Nonlinear Anal. 71, 5269–5300 (2009)

    Google Scholar 

  15. Diagana, T., N’Guérékata, G.M., Van Minh, N.: Almost automorphic solutions of evolution equations. Proc. Am. Math. Soc. 132(11), 3289–3298 (2004)

    Article  MATH  Google Scholar 

  16. Furumochi, T., Naito, T., Van Minh, N.: Boundedness and almost periodicity of solutions of partial functional differential equations. J. Differ. Equ. 180, 125–152 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

  18. Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien (1997)

    Google Scholar 

  19. Hilfer, R.: Applications of Fractional Calculus Physics. World Scientific, River Edge (2000)

    Book  MATH  Google Scholar 

  20. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science, Amsterdam (2006)

    Book  MATH  Google Scholar 

  21. Lang, S.: Complex Analysis. Graduate texts in Math., vol. 103. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  22. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Lizama, C.: An operator theoretical approach to a class of fractional order differential equations. Appl. Math. Lett. 24(2), 184–190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lizama, C., N’Guérékata, G.M.: Bounded mild solutions for semilinear integro-differential equations in Banach spaces. Integral Equ. Oper. Theory 68(2), 207–227 (2010)

    Article  MATH  Google Scholar 

  26. Lizama, C., Poblete, F.: On a functional equation associated with (a,k)-regularized resolvent families. Abstr. Appl. Anal. 2012, 495487 (2012). doi:10.1155/2012/495487

    MathSciNet  Google Scholar 

  27. Lizama, C., Ponce, R.: Bounded solutions to a class of semilinear integro-differential equations in Banach spaces. Nonlinear Anal. 74, 3397–3406 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010)

    Book  Google Scholar 

  29. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  30. Mishura, S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  31. Mophou, G., N’Guérékata, G.M.: On some classes of almost automorphic functions and applications to fractional differential equations. Comput. Math. Appl. 59, 1310–1317 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. N’Guérékata, G.M.: Almost Automorphic and Almost Periodic Functions Abstract Spaces. Kluwer Academic/Plenum, New York (2001)

    Book  MATH  Google Scholar 

  33. 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  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  35. N’Guérékata, G.M.: Almost automorphic solutions to second-order semilinear evolution equations. Nonlinear Anal. 71, e432–e435 (2009)

    Article  MATH  Google Scholar 

  36. Nunziato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–304 (1971)

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  38. Saxena, R., Srivastava, H.: Some Volterra-type fractional integro-differential equations with a multivariable confluent hypergeometric function as their kernel. J. Integral Equ. Appl. 17(2), 199–217 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

    Rodrigo Ponce

Authors
  1. Rodrigo Ponce
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rodrigo Ponce.

Additional information

Communicated by Abdelaziz Rhandi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ponce, R. Bounded mild solutions to fractional integro-differential equations in Banach spaces. Semigroup Forum 87, 377–392 (2013). https://doi.org/10.1007/s00233-013-9474-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-013-9474-y

Keywords

Search

Navigation