Abstract and Applied Analysis

Positive Solutions for the Eigenvalue Problem of Semipositone Fractional Order Differential Equation with Multipoint Boundary Conditions

Ge Dong

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Abstract

We study the existence of positive solution for the eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions by using known Krasnosel'skii's fixed point theorem. Some sufficient conditions that guarantee the existence of at least one positive solution for eigenvalues   λ > 0 sufficiently small and λ > 0 sufficiently large are established.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 925010, 9 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412279736

Digital Object Identifier
doi:10.1155/2014/925010

Mathematical Reviews number (MathSciNet)
MR3198274

Citation

Dong, Ge. Positive Solutions for the Eigenvalue Problem of Semipositone Fractional Order Differential Equation with Multipoint Boundary Conditions. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 925010, 9 pages. doi:10.1155/2014/925010. https://projecteuclid.org/euclid.aaa/1412279736


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References

  • H. Feng and D. Bai, “Existence of positive solutions for semipositone multi-point boundary value problems,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2287–2292, 2011.
  • C. Yuan, “Multiple positive solutions for $(n-1,1)$-type semipositone conjugate boundary value problems of nonlinear fractional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, no. 36, pp. 1–12, 2010.
  • X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1420–1433, 2012.
  • C. S. Goodrich, “Positive solutions to boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 1, pp. 417–432, 2012.
  • C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1251–1268, 2011.
  • X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012.
  • C. Yang and J. Yan, “Positive solutions for third-order Sturm-Liouville boundary value problems with p-Laplacian,” Computers and Mathematics with Applications, vol. 59, no. 6, pp. 2059–2066, 2010.
  • J. Wang, H. Xiang, and Z. Liu, “Positive solutions for three-point boundary value problems of nonlinear fractional differential equations with $p$-Laplacian,” Far East Journal of Applied Mathematics, vol. 37, no. 1, pp. 33–47, 2009.
  • J. Wang and H. Xiang, “Upper and lower solutions method for a class of singular fractional boundary value problems with $p$-Laplacian operator,” Abstract and Applied Analysis, vol. 2010, Article ID 971824, 12 pages, 2010.
  • G. Chai, “Positive solutions for boundary value problem of fractional differential equation with $p$-Laplacian operator,” Boundary Value Problems, vol. 2012, article 18, 2012.
  • X. Zhang, L. Liu, and Y. Wu, “The uniqueness of positive solution for a singular fractional differential system involving derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 6, pp. 1400–1409, 2013.
  • J. R. L. Webb, “Nonlocal conjugate type boundary value problems of higher order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1933–1940, 2009.
  • X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555–560, 2012.
  • 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.
  • X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012.
  • J. J. Nieto and J. Pimentel, “Positive solutions of a fractional thermostat model,” Boundary Value Problems, vol. 2013, article 5, 2013.
  • Y. Li and S. Lin, “Positive solution for the nonlinear Hadamard type fractional differential equation with $p$-Laplacian,” Journal of Function Spaces and Applications, vol. 2013, Article ID 951643, 10 pages, 2013.
  • X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4680–4691, 2013.
  • A. A. M. Arafa, S. Z. Rida, and M. Khalil, “Fractional modeling dynamics of HIV and CD4$^{+}$ T-cells during primary infection,” Nonlinear Biomedical Physics, vol. 6, no. 1, article 1, 2012.
  • X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012.
  • 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.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
  • 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. \endinput