Skip to main content
SpringerLink
Log in

Fractional boundary value problems with singularities in space variables

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We are concerned with the existence of solutions for the singular fractional boundary value problem \(^{c}\kern-1pt D^{\alpha}u = f(t,u)\), u(0)+u(1)=0, u′(0)=0, where α∈(1,2), f∈C([0,1]×(ℝ∖{0})) and lim x→0 f(t,x)=∞ for all t∈[0,1]. Here, \(^{c}\kern-1pt D\) is the Caputo fractional derivative. Increasing solutions of the problem vanish at points of (0,1), that is, they “pass through” the singularity of f inside of (0,1). The results are based on combining regularization and sequential techniques with a nonlinear alternative. In limit processes, the Vitali convergence theorem is used.

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. Podlubny, I.: Fractional differential equations. In: Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)

    Google Scholar 

  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  3. Diethelm, K.: The Analysis of Fractional Differential Equations. Lectures Notes in Mathematics. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  4. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos, vol. 3. World Scientific, Singapore (2012)

    MATH  Google Scholar 

  5. Abbas, S., Banerjee, M., Momani, S.: Dynamical analysis of fractional-order modified logistic model. Comput. Math. Appl. 62, 1098–1104 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Daftardar-Gejji, V., Bhalekar, S.: Boundary value problems for multi-term fractional differential equations. J. Math. Anal. Appl. 345, 754–765 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dalir, M., Bashour, M.: Applications of fractional calculus. Appl. Math. Sci. 4(21), 1021–1032 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results and problems I. Appl. Anal. 78(1–2), 153–192 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results and problems II. Appl. Anal. 78(2), 435–493 (2002)

    Article  MathSciNet  Google Scholar 

  10. Agarwal, RP., Ahmad, B.: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, 1200–1214 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, 916–924 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caballero, J., Harjani, J., Sadarangani, K.: Existence and uniqueness of positive solution for a boundary value problem of fractional order. Abstr. Appl. Anal. (2011). doi:10.1155/2011/165641

    Google Scholar 

  13. Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71, 2391–2396 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo, Y.: Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations. Bull. Korean Math. Soc. 47(1), 81–87 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang, D., Yuan, C.: The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. 72, 710–719 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, C.F., Luo, X.N., Zhou, Y.: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 1363–1375 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin, X., Sun, W., Jiang, D.: Existence and uniqueness of solutions for boundary value problems to the singular one-dimensional p-Laplacian. Bound. Value Probl. (2008). doi:10.1155/2008/194234

    MathSciNet  Google Scholar 

  18. Qiu, T., Bai, Z.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Nonlinear Sci. Appl. 1(3), 123–131 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Rehman, M., Khan, R.A., Asif, N.A.: Three point boundary value problems for nonlinear fractional differential equations. Acta Math. Sci. 31, 1337–1346 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Wang, J., Xiang, H., Liu, Z.G.: Existence of concave positive solutions for boundary value problem of nonlinear fractional differential equation with p-Laplacian operator. Int. J. Math. Math. Sci. (2010). doi:10.1155/2010/495138

    MathSciNet  Google Scholar 

  21. Wang, J., Xiang, H.: Upper and lower solutions method for a class of singular fractional boundary value problems with p-Laplacian operator. Abstr. Appl. Anal. (2010). doi:10.1155/2010/971824

    MathSciNet  Google Scholar 

  22. Wang, J., Xiang, H., Zhao, Y.: Monotone and concave positive solutions to a boundary value problem for higher-order fractional differential equation. Abstract and Applied Analysis (2011). doi:10.1155/2011/430457

  23. Arara, A., Benchohra, M., Hamidi, N., Nieto, J.J.: Fractional order differential equations on an unbounded domain. Nonlinear Anal. 72, 580–586 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Baleanu, D., Mustafa, O.G., Agarwal, R.P.: An existence result for a superlinear fractional differential equation. Appl. Math. Lett. 23, 1129–1132 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhao, X., Ge, W.: Unbounded solutions for a fractional boundary value problems on the infinite interval. Acta Appl. Math. 109, 495–505 (2010). doi:10.1007/s10440-008-9329-9

    Article  MathSciNet  MATH  Google Scholar 

  26. Agarwal, R.P., O’Regan, D., Staněk, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57–68 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Staněk, S.: The existence of positive solutions of singular fractional boundary value problems. Comput. Math. Appl. 62, 1379–1388 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Agarwal, R.P., O’Regan, D., Staněk, S.: Positive solutions for mixed problems of singular fractional differential equations. Math. Nachr. 285(1), 27–41 (2012). doi:10.1002/mana.201000043

    Article  MathSciNet  MATH  Google Scholar 

  29. Cui, Y.: Existence results for singular boundary value problem of nonlinear fractional differential equation. Abstr. Appl. Anal. (2011). doi:10.1155/2011/605614

    Google Scholar 

  30. Xu, X., Jiang, D., Yuan, C.: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal. 71, 4676–4688 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yuan, C., Jiang, D., Xu, X.: Singular positone and semipositone boundary value problems of nonlinear fractional differential equations. Math. Probl. Eng. (2009). doi:10.1155/2009/535209

    MathSciNet  Google Scholar 

  32. Zhang, S.: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 59, 1300–1309 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rachůnková, I., Staněk, S., Tvrdý, M.: Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations. In: Cañada, A., Drábek, P., Fonda, A. (eds.) Handbook of Differential Equations. Ordinary Differential Equations, vol. 3, pp. 607–723. Elsevier, Amsterdam (2006)

    Google Scholar 

  34. Rachůnková, I., Staněk, S., Tvrdý, M.: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations. Hindawi, New York (2009)

    Book  Google Scholar 

  35. Bartle, R.G.: A Modern Theory of Integration. AMS, Providence, Rhode Island (2001)

    MATH  Google Scholar 

  36. Natanson, I.P.: Theorie der Funktionen Einer Reellen Veränderlichen, Herausgegeben Von Karl Bögel, 4. Aufl. (German). Mathematische Lehrbücher und Monographien. Akademie, Berlin (1975). English transl.: Theory of Functions of Real Variable. Ungar, New York (1955)

    MATH  Google Scholar 

  37. Agarwal, R.P., O’Regan, D., Staněk, S.: Positive solutions of singular boundary value problems for delay differential equations. Dyn. Syst. Appl. 16, 755–770 (2007)

    MATH  Google Scholar 

  38. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The research of the second author was supported by grants PrF-2011-022 and PrF-2012-017.

Author information

Authors and Affiliations

  1. Department of Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

  2. Department of Mathematical Analysis, Faculty of Science, Palacký University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Svatoslav Staněk

Authors
  1. Donal O’Regan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Svatoslav Staněk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Svatoslav Staněk.

Rights and permissions

Reprints and permissions

About this article

Cite this article

O’Regan, D., Staněk, S. Fractional boundary value problems with singularities in space variables. Nonlinear Dyn 71, 641–652 (2013). https://doi.org/10.1007/s11071-012-0443-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0443-x

Keywords

Search

Navigation