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Abstract and Applied Analysis
Volume 2013, Article ID 579740, 20 pages
http://dx.doi.org/10.1155/2013/579740
Research Article

Positive Solutions to Fractional Boundary Value Problems with Nonlinear Boundary Conditions

1Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran
2Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, 06530 Ankara, Turkey
3Institute of Space Sciences, P.O. BOX MG-23, Magurele, 76900 Bucharest, Romania
4Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

Received 20 March 2013; Accepted 5 April 2013

Academic Editor: Juan J. Trujillo

Copyright © 2013 Nemat Nyamoradi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advances in Fractional Calculus, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. II,” Applicable Analysis, vol. 81, no. 2, pp. 435–493, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 434–442, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  8. D. Băleanu, O. G. Mustafa, and R. P. Agarwal, “On the solution set for a class of sequential fractional differential equations,” Journal of Physics A, vol. 43, no. 38, Article ID 385209, 7 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. D. Baleanu, H. Mohammadi, and Sh. Rezapour, “Some existence results on nonlinear fractional differential equations,” Philosophical Transactions of the Royal Society A, vol. 371, no. 1990, 2013. View at Publisher · View at Google Scholar
  10. H. Scher and E. W. Montroll, “Anomalous transit-time dispersion in amorphous solids,” Physical Review B, vol. 12, no. 6, pp. 2455–2477, 1975. View at Publisher · View at Google Scholar
  11. K. Diethelm and A. D. Freed, “On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity,” in Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, F. Keil, W. Mackens, H. voss, and J. Werther, Eds., Springer, Heidelberg, Germany, 1999. View at Google Scholar
  12. L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, pp. 81–88, 1991. View at Publisher · View at Google Scholar
  13. W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-semilar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995. View at Google Scholar
  14. F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 291–348, Springer, New York, NY, USA, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” Journal of Chemical Physics, vol. 103, pp. 7180–7186, 1995. View at Google Scholar
  16. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. H. A. H. Salem, “On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 565–572, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  18. L. S. Leibenson, “General problem of the movement of a compressible fluid in a porous medium,” Izvestiia Akademii Nauk Kirgizskoi SSR, vol. 9, pp. 7–10, 1983. View at Google Scholar
  19. B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 599–606, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. G. Dai, “Bifurcation and nodal solutions for p-Laplacian problems with non-asymptotic nonlinearity at 0 or ∞,” Applied Mathematics Letters, vol. 26, no. 1, pp. 46–50, 2013. View at Google Scholar
  21. T. Chen, W. Liu, and C. Yang, “Antiperiodic solutions for Liénard-type differential equation with p-Laplacian operator,” Boundary Value Problems, vol. 2010, Article ID 194824, 12 pages, 2010. View at Google Scholar · View at MathSciNet
  22. D. Jiang and W. Gao, “Upper and lower solution method and a singular boundary value problem for the one-dimensional p-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 631–648, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  23. L. F. Lian and W. G. Ge, “The existence of solutions of m-point p-Laplacian boundary value problems at resonance,” Acta Mathematicae Applicatae Sinica, vol. 28, no. 2, pp. 288–295, 2005. View at Google Scholar · View at MathSciNet
  24. B. Liu and J. S. Yu, “Existence of solutions for the periodic boundary value problems with p-Laplacian operator,” Journal of Systems Science and Mathematical Sciences, vol. 23, no. 1, pp. 76–85, 2003. View at Google Scholar · View at MathSciNet
  25. J. J. Zhang, W. B. Liu, J. B. Ni, and T. Y. Chen, “Multiple periodic solutions of p-Laplacian equation with one-side Nagumo condition,” Journal of the Korean Mathematical Society, vol. 45, no. 6, pp. 1549–1559, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  26. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1993. View at MathSciNet
  27. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  28. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  29. C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1363–1375, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan, New York, NY, USA, 1964. View at MathSciNet