Abstract and Applied Analysis

Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation

Moustafa El-Shahed

Full-text: Open access

Abstract

We are concerned with the existence and nonexistence of positive solutions for the nonlinear fractional boundary value problem: D 0 + α u ( t ) + λ a ( t )  f ( u ( t ) ) = 0 ,  0 < t < 1 ,  u ( 0 ) = u ( 0 ) = u ( 1 ) = 0 , where 2 < α < 3 is a real number and D 0 + α is the standard Riemann-Liouville fractional derivative. Our analysis relies on Krasnoselskiis fixed point theorem of cone preserving operators. An example is also given to illustrate the main results.

Article information

Source
Abstr. Appl. Anal., Volume 2007 (2007), Article ID 10368, 8 pages.

Dates
First available in Project Euclid: 27 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1204126601

Digital Object Identifier
doi:10.1155/2007/10368

Mathematical Reviews number (MathSciNet)
MR2353784

Zentralblatt MATH identifier
1149.26012

Citation

El-Shahed, Moustafa. Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation. Abstr. Appl. Anal. 2007 (2007), Article ID 10368, 8 pages. doi:10.1155/2007/10368. https://projecteuclid.org/euclid.aaa/1204126601


Export citation

References

  • D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.
  • H. Wang, ``On the existence of positive solutions for semilinear elliptic equations in the annulus,'' Journal of Differential Equations, vol. 109, no. 1, pp. 1--7, 1994.
  • Z. Bai and H. Lü, ``Positive solutions for boundary value problem of nonlinear fractional differential equation,'' Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495--505, 2005.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • S. Zhang, ``Existence of solution for a boundary value problem of fractional order,'' Acta Mathematica Scientia, vol. 26, no. 2, pp. 220--228, 2006.
  • S. Zhang, ``Positive solutions for boundary-value problems of nonlinear fractional differential equations,'' Electronic Journal of Differential Equations, vol. 2006, no. 36, pp. 1--12, 2006.
  • H. Amann, ``Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,'' SIAM Review, vol. 18, no. 4, pp. 620--709, 1976.
  • M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, The Netherlands, 1964.
  • H.-R. Sun and W.-K. Wen, ``On the number of positive solutions for a nonlinear third order boundary value problem,'' International Journal of Difference Equations, vol. 1, no. 1, pp. 165--176, 2006.