Table of Contents Author Guidelines Submit a Manuscript
Complexity
Volume 2017, Article ID 7676814, 9 pages
https://doi.org/10.1155/2017/7676814
Research Article

Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations

1Department of Mathematics, University of Malakand, Dir (L), Khyber Pakhtunkhwa, Pakistan
2Department of Mathematics, Sun Yat-Sen University, Guangzhou, China

Correspondence should be addressed to Yongjin Li; nc.ude.usys.liam@jylsts

Received 15 March 2017; Revised 26 April 2017; Accepted 11 May 2017; Published 20 July 2017

Academic Editor: Sundarapandian Vaidyanathan

Copyright © 2017 Muhammad Iqbal 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. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of New York, NY, USA, Elsevier, 2006. View at MathSciNet
  2. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  3. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  4. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. View at MathSciNet
  5. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. R. A. Khan and K. Shah, “Existence and uniqueness of solutions to fractional order multi-point boundary value problems,” Communication in Applied Analysis, vol. 19, pp. 515–526, 2015. View at Google Scholar
  7. X. Wang, L. Wang, and Q. Zeng, “Fractional differential equations with integral boundary conditions,” Journal of Nonlinear Science and Its Applications, vol. 8, no. 4, pp. 309–314, 2015. View at Google Scholar · View at MathSciNet
  8. K. Shah, H. Khalil, and R. A. Khan, “Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations,” Chaos, Solitons & Fractals, vol. 77, pp. 240–246, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. A. Nanware and D. B. Dhaigude, “Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions,” Journal of Nonlinear Science and its Applications. JNSA, vol. 7, no. 4, pp. 246–254, 2014. View at Google Scholar · View at MathSciNet
  10. L. Lv, J. Wang, and W. Wei, “Existence and uniqueness results for fractional differential equations with boundary value conditions,” Opuscula Mathematica, vol. 31, no. 4, pp. 629–643, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. B. C. Dhage, “Fixed point theorems in ordered Banach algebras and applications,” Panamerican Mathematical Journal, vol. 9, no. 4, pp. 93–102, 1999. View at Google Scholar · View at MathSciNet
  12. M. A. Herzallah and D. Baleanu, “On fractional order hybrid differential equations,” Abstract and Applied Analysis, vol. 2014, Article ID 389386, 7 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  13. T. Bashiri, S. M. Vaezpour, and C. Park, Fixed Point Theory and Application to Fractional Hybrid Differential Problems, Springer, 2016.
  14. B. C. Dhage, “A fixed point theorem in Banach algebras with applications to fractional integral equations,” Kyungpook Mathematical Journal, vol. 44, pp. 145–155, 2004. View at Google Scholar · View at MathSciNet
  15. B. C. Dhage and S. B. Dhage, “Approximating solutions of nonlinear PBVPs of second-order differential equations via hybrid fixed point theory,” Electronic Journal of Differential Equations, vol. 20, pp. 1–10, 2015. View at Google Scholar · View at MathSciNet
  16. B. C. Dhage and V. Lakshmikantham, “Basic results on hybrid differential equations,” Nonlinear Analysis. Hybrid Systems, vol. 4, no. 3, pp. 414–424, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. B. C. Dhage and N. S. Jadhav, “Basic results in the theory of hybrid differential equations with linear perturbations of second type,” Tamkang Journal of Mathematics, vol. 44, no. 2, pp. 171–186, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. H. Lu, S. Sun, D. Yang, and H. Teng, “Theory of fractional hybrid differential equations with linear perturbations of second type,” Boundary Value Problems, vol. 2013, no. 23, 16 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. T. A. Burton, “A fixed-point theorem of Krasnoselskii,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 11, no. 1, pp. 85–88, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  20. D. O'Regan, Y. J. Cho, and Y.-Q. Chen, Topological Degree Theory and Applications, vol. 10 of Series in Mathematical Analysis and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, vol. 40 of NSFCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1979. View at MathSciNet
  22. G. Dinca, P. Jebelean, and J. Mawhin, “Variational and topological methods for Dirichlet problems with p-Laplacian,” Portugaliae Mathematica, vol. 58, no. 3, pp. 339–378, 2001. View at Google Scholar · View at MathSciNet
  23. F. Isaia, “On a nonlinear integral equation without compactness,” Acta Mathematica Universitatis Comenianae. New Series, vol. 75, no. 2, pp. 233–240, 2006. View at Google Scholar · View at MathSciNet
  24. J. Wang, Y. Zhou, and W. Wei, “Study in fractional differential equations by means of topological degree methods,” Numerical Functional Analysis and Optimization. An International Journal, vol. 33, no. 2, pp. 216–238, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. A. Ali, B. Samet, K. Shah, and R. Khan, “Existence and stability of solution to a toppled systems of differential equations of non-integer order,” Boundary Value Problems, vol. 2017, 16 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  26. K. Shah and R. A. Khan, “Existence and uniqueness results to a coupled system of fractional order boundary value problems by topological degree theory,” Numerical Functional Analysis and Optimization. An International Journal, vol. 37, no. 7, pp. 887–899, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. K. Shah, A. Ali, and R. A. Khan, “Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems,” Boundary Value Problems, vol. 2016, no. 43, 12 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  28. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  29. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  30. V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, '' Cambridge Academic Publishers, Cambridge, UK, 2009.
  31. S. S. Chang, Y. J. Cho, and N. J. Huang, “Coupled fixed point theorems with applications,” Journal of the Korean Mathematical Society, vol. 33, no. 3, pp. 575–585, 1996. View at Google Scholar · View at MathSciNet