Skip to main content
Log in

Solvability of boundary-value problems for nonlinear fractional differential equations

  • Published:
Ukrainian Mathematical Journal Aims and scope

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations

$$ \begin{array}{*{20}{c}} {{{\mathbf{D}}^\alpha }u(t) + {{\lambda }}\left[ {f\left( {t,u(t)} \right) + q(t)} \right] = 0,\quad 0 < t < 1,} \\ {u(0) = 0,\quad u(1) = \beta u(\eta ),} \\ \end{array}$$

where λ > 0 is a parameter, 1 < α ≤ 2, η ∈ (0, 1), \(\beta \in \mathbb{R} = \left({-\infty, +\infty} \right) \), βη α−1 ≠ 1, D α is a Riemann–Liouville differential operator of order α, \(f:\left(0, 1 \right) \times \mathbb{R} \to \mathbb{R} \) is continuous, f may be singular for t = 0 and/or t = 1, and q(t) : [0, 1] → [0, +∞) We give some sufficient conditions for the existence of nontrivial solutions to the formulated boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. In particular, we do not use the assumption of nonnegativity and monotonicity of f essential for the technique used in almost all available literature.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. O. P. Agrawal, “Formulation of Euler–Lagrange equations for fractional variational problems,” J. Math. Anal. Appl., 272, 368–379 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” J. Math. Appl., 204, 609–625 (1996).

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  4. B. Liu, “Positive solutions of a nonlinear three-point boundary-value problem,” Comput. Math. Appl., 44, 201–211 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Podlubny, “Fractional differential equations,” Math. Sci. and Eng., 198 (1999).

  6. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives (Theory and Applications), Gordon & Breach (1993).

  7. Zhanbing Bai and Haishen Lü, “Positive solutions for boundary-value problem of nonlinear fractional differential equation,” J. Math. Anal. Appl., 311, 495–505 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  8. Shu-qin Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” J. Math. Anal. Appl., 252, 804–812 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  9. Shu-qin Zhang, “Existence of positive solution for some class of nonlinear fractional differential equations,” J. Math. Anal. Appl., 278, No. 1, 136–148 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  10. E. R. Kaufmann and E. Mboumi, “Positive solutions of boundary-value problems for nonlinear fractional differential equations,” Electron. J. Qual. Theory Different. Equat., No. 3, 1–11 (2008).

  11. A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results, and problems. II,” Appl. Anal., 81, 435–493 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. M. Nakhushev, “Sturm–Liouville problem for a second-order ordinary differential equation with fractional derivatives in the lower terms,” Dokl. Akad. Nauk SSSR, 234, 308–311 (1977).

    MathSciNet  Google Scholar 

  13. A. H. Salem Hussein, “On the fractional order m-point boundary-value problem in reflexive Banach spaces and weak topologies,” J. Comput. Appl. Math., 224, 565–572 (2009).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 9, pp. 1211–1219, September, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, Y. Solvability of boundary-value problems for nonlinear fractional differential equations. Ukr Math J 62, 1409–1419 (2011). https://doi.org/10.1007/s11253-011-0439-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-011-0439-6

Keywords

Navigation