Abstract and Applied Analysis

Existence of Positive Solution for Semipositone Fractional Differential Equations Involving Riemann-Stieltjes Integral Conditions

Wei Wang and Li Huang

Full-text: Open access

Abstract

The existence of at least one positive solution is established for a class of semipositone fractional differential equations with Riemann-Stieltjes integral boundary condition. The technical approach is mainly based on the fixed-point theory in a cone.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 723507, 17 pages.

Dates
First available in Project Euclid: 4 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365099925

Digital Object Identifier
doi:10.1155/2012/723507

Mathematical Reviews number (MathSciNet)
MR2965475

Zentralblatt MATH identifier
1246.34013

Citation

Wang, Wei; Huang, Li. Existence of Positive Solution for Semipositone Fractional Differential Equations Involving Riemann-Stieltjes Integral Conditions. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 723507, 17 pages. doi:10.1155/2012/723507. https://projecteuclid.org/euclid.aaa/1365099925


Export citation

References

  • W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach of self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.
  • F. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmache, “Relaxation in filled polymers: a fractional calculus approach,” Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995.
  • I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, London, UK, 1999.
  • I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  • V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and derivatives Theory and Applications, Gordon and Breach Science Publishers, Yverdon-les-Bains, Switzerland, 1993.
  • R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
  • 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.
  • 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 Article ID 512127, 16 pages, 2012.
  • 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.
  • Y. Wang, L. Liu, and Y. Wu, “Positive solutions of a fractional boundary value problem with changing sign nonlinearity,” Abstract and Applied Analysis, vol. 2012, Article ID 149849, 12 pages, 2012.
  • X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher orderperturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, http://dx.doi.org/10.1016/j.amc.2012.07.046.
  • J. R. L. Webb and G. Infante, “Positive solutions of nonlocal boundary value problems involving integral conditions,” Nonlinear Differential Equations and Applications, vol. 15, no. 1-2, pp. 45–67, 2008.
  • J. R. L. Webb and G. Infante, “Positive solutions of nonlocal boundary value problems: a unified approach,” Journal of the London Mathematical Society, vol. 74, no. 3, pp. 673–693, 2006.
  • 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. Hao, L. Liu, Y. Wu, and Q. Sun, “Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 6, pp. 1653–1662, 2010.
  • S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1300–1309, 2010.
  • C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1050–1055, 2010.
  • M. Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1038–1044, 2010.
  • Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a nonlocal fractional differential equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 11, pp. 3599–3605, 2011.
  • 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.
  • R. Aris, Introduction to the Analysis of Chemical Reactors, Prentice Hall, Englewood Cliffs, NJ, USA, 1965.
  • X. Zhang and L. Liu, “Positive solutions of superlinear semipositone singular Dirichlet boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 525–537, 2006.
  • Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6434–6441, 2011.
  • D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, NY, USA, 1988.