Skip to main content
Log in

Unique positive solution for a fractional boundary value problem

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

In this work we consider the unique positive solution for the following fractional boundary value problem

$\left\{ \begin{gathered} D_{0 + }^\alpha u(t) = - f(t,u(t)),t \in [0,1], \hfill \\ u(0) = u'(0) = u'(1) = 0. \hfill \\ \end{gathered} \right. $

Here α ∈ (2, 3] is a real number, D α0+ is the standard Riemann-Liouville fractional derivative of order α. By using the method of upper and lower solutions and monotone iterative technique, we also obtain that there exists a sequence of iterations uniformly converges to the unique solution.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. C. Bai, Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 384, No 2 (2011), 211–231.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. El-Shahed, Positive solutions for boundary value problems of nonlinear fractional differential equation. Abs. Appl. Anal. Volume 2007, Article ID 10368.

    Google Scholar 

  3. D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones. Academic Press, Orlando (1988).

    MATH  Google Scholar 

  4. D. Guo, Positive solutions of Hammerstein integral equations of polynomial type with applications. Chinese Ann. Math. Ser. A. 4, No 5 (1983), 645–656.

    MathSciNet  MATH  Google Scholar 

  5. J. Graef, L. Kong, B. Yang, Positive solutions for a semipositone fractional boundary value problem with a forcing term. Fract. Calc. Appl. Anal. 15, No 1 (2012), 8–24; DOI: 10.2478/s13540-012-0002-7; http://link.springer.com/article/10.2478/s13540-012-0002-7.

    MathSciNet  MATH  Google Scholar 

  6. J. Graef, L. Kong, Q. Kong, M. Wang, Uniqueness of positive solutions of fractional boundary value problems with nonhomogeneous integral boundary conditions. Fract. Calc. Appl. Anal. 15, No 3 (2012), 509–528; DOI: 10.2478/s13540-012-0036-x; http://link.springer.com/article/10.2478/s13540-012-0036-x.

    MathSciNet  MATH  Google Scholar 

  7. M. Krasnoselski, P. Zabreiko, Geometrical Methods of Nonlinear Analysis. Springer (1984).

    Book  Google Scholar 

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

    Book  MATH  Google Scholar 

  9. V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009).

    MATH  Google Scholar 

  10. S. Liang, J. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 71, No 11 (2009), 5545–5550.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. R. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem. In: Fixed Point Theory, Lecture Notes in Mathematics 886 (1981), Springer, 309–330.

    Chapter  Google Scholar 

  13. I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering 198, Academic Press, San Diego — CA (1999).

    MATH  Google Scholar 

  14. S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993).

    MATH  Google Scholar 

  15. Z. Wei, Q. Li, J. Che, Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 367, No 1 (2010), 260–272.

    Article  MathSciNet  MATH  Google Scholar 

  16. Z. Wei, W. Dong, J. Che, Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. Nonlinear Anal. 73, No 10 (2010), 3232–3238.

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Zhao, S. Sun, Z. Han, Q. Li, The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, No 4 (2011), 2086–2097.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Keyu Zhang.

About this article

Cite this article

Zhang, K., Xu, J. Unique positive solution for a fractional boundary value problem. fcaa 16, 937–948 (2013). https://doi.org/10.2478/s13540-013-0057-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s13540-013-0057-0

MSC 2010

Key Words and Phrases

Navigation