Skip to main content
SpringerLink
Log in

Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we study boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces. A generalized singular type Gronwall inequality is given to obtain an important priori bounds. Some sufficient conditions for the existence of solutions are established by virtue of fractional calculus and fixed point method under some mild conditions. Two examples are given to illustrate the results.

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. Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley-Interscience, New York (1984)

    MATH  Google Scholar 

  3. Balachandran, K., Park, J.Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Anal. 71, 4471–4475 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Balachandran, K., Kiruthika, S., Trujillo, J.J.: Existence results for fractional impulsive integrodifferential equations in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 16, 1970–1977 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008)

    MATH  MathSciNet  Google Scholar 

  6. El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433–440 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)

    MATH  Google Scholar 

  8. Henderson, J., Ouahab, A.: Fractional functional differential inclusions with finite delay. Nonlinear Anal. 70, 2091–2105 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hernández, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 73, 3462–3471 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

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

    MATH  Google Scholar 

  12. Mophou, G.M., N’Guérékata, G.M.: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl. Math. Comput. 216, 61–69 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

    MATH  Google Scholar 

  14. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  15. Smart, D.R.: Fixed Point Theorems, vol. 66. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  16. Wang, J.R., Xiang, X., Wei, W., Chen, Q.: The generalized Gronwall inequality and its application to periodic solutions of integrodifferential impulsive periodic system on Banach space. J. Inequal. Appl. 2008, 1–22 (2008).

    MATH  MathSciNet  Google Scholar 

  17. Wang, J.R., Wei, W., Yang, Y.: Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control. Opusc. Math. 30, 217–234 (2010)

    MathSciNet  Google Scholar 

  18. Wang, J.R., Yang, Y., Wei, W.: Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces. Opusc. Math. 30, 361–381 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Wang, J.R., Wei, W., Yang, Y.: On some impulsive fractional differential equations in Banach spaces. Opusc. Math. 30, 507–525 (2010)

    MATH  MathSciNet  Google Scholar 

  20. Wang, J.R., Wei, W., Yang, Y.: Fractional nonlocal integrodifferential equations and its optimal control in Banach spaces. J. KSIAM 14, 79–91 (2010)

    MathSciNet  Google Scholar 

  21. Wang, J.R., Zhou, Y.: Time optimal control problem of a class of fractional distributed systems. Int. J. Dyn. Differ. Equ. 3, 363–382 (2010)

    Google Scholar 

  22. Wang, J.R., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. 12, 262–272 (2011)

    Article  MATH  Google Scholar 

  23. Zhou, Y.: Existence and uniqueness of fractional functional differential equations with unbounded delay. Int. J. Dyn. Differ. Equ. 1, 239–244 (2008)

    MATH  Google Scholar 

  24. Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. 71, 3249–3256 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Zhou, Y., Jiao, F.: Existence of extremal solutions for discontinuous fractional functional differential equations. Int. J. Dyn. Differ. Equ. 2, 237–252 (2009)

    MATH  MathSciNet  Google Scholar 

  26. Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010)

    MATH  MathSciNet  Google Scholar 

  27. Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 11, 4465–4475 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, P.R. China

    JinRong Wang & Linli Lv

  2. Department of Mathematics, Xiangtan University, Xiangtan, 411105, P.R. China

    Yong Zhou

Authors
  1. JinRong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Linli Lv
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JinRong Wang.

Additional information

This work is supported by Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11026102), National Natural Science Foundation of China (Grant Nos. 10971173, 10961009), National Natural Science Foundation of Guizhou Province (2010, No. 2142).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J., Lv, L. & Zhou, Y. Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces. J. Appl. Math. Comput. 38, 209–224 (2012). https://doi.org/10.1007/s12190-011-0474-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-011-0474-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation