Title: Existence of Multiple Positive Solutions for the System of Nonlinear Fractional Order Boundary Value Problem
Author(s): Sabbavarapu Nageswara Rao
Pages: 81-92
Cite as:
Sabbavarapu Nageswara Rao, Existence of Multiple Positive Solutions for the System of Nonlinear Fractional Order Boundary Value Problem, Int. J. Anal. Appl., 11 (2) (2016), 81-92.

Abstract


This paper is concerned with boundary value problems for system of nonlinear fractional differential equations involving the Caputo fractional derivatives. Under the suitable conditions, the existence and multiplicity of positive solutions are established by using abstract fixed point theorems.

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References


  1. A. Y. Al-Hossain, Eigenvalues for iterative systems of nonlinear Caputo fractionalorder three point boundary value problems, J. Appl. Math. Comput. (2015), DOI 10.1007/s12190-015-0935-1.

  2. C. Bai and W. Sun, Existence and multiplicity of positive solutions for singular fractional boundary value problems, Comput. Math. Appl. 63 (2012), 1369-1381.

  3. C. Bai, W. Sun and W. Zhang, Positive solutions for boundary value problems of a singular fractional differential equations, Abstr. Appl. Anal. 2013 (2013), article ID 129640.

  4. Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), 495-505.

  5. G. Chai, Existence results of positive solutions for boundary value problems of fractional differential equations, Bound. Value Probl. 2013 (2013), Art. ID 109.

  6. K. Deimling, Nonlinear Functional Analysis, Springer-verlag, Berlin, 1985.

  7. P. W. Eloe and J. Henderson, Positive solutions for higher order ordinary differential equations, Electron. J. Differential Equations. 3 (1995), 1-8.

  8. L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), 743-748.

  9. D. J. Guo and L. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988.

  10. CS. Goodrich, On a fractional boundary value problem with fractional boundary conditions, Appl. Math. Lett. 25 (2012), 1101-1105.

  11. L. Hu and L. Wang, Multiple positive solutions of boundary value problems for systems of nonlinear second-order differential equations, J. Math. Anal. Appl. 335 (2007), 1052-1060.

  12. J. Henderson and R. Luca, Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems, Nonlinear Differential Equations Appl. 20 (2013), No. 3, 1035-1054.

  13. A. Kameswarao and S. Nageswararao, Multiple positive solutions of boundary value problems for systems of nonlinear second-order dynamic equations on time scales, Math. Commun. 15 (2010), No. 1, 129-138.

  14. M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.

  15. V. Lakshmikanthan, Theory of fractional differential equations, Nonlinear Anal. 69 (2008) 3337-3343.

  16. V. Lakshmikanthan, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009).

  17. S. Liang and J. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Nonlinear Anal. 71 (2009), 5545-5550.

  18. K. S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993).

  19. S. Nageswararao, Multiple positive solutions for a system of Riemann-Liouville fractional order two-point boundary value problems, Panamer. Math. J. 25 (2015), No. 1, 66-81.

  20. S. Nageswararao, Existence of positive solutions for Riemann-Liouville fractional order three-point boundary value problem, Asian-Eur. J. Math. 8 (2015), No. 4, Art. ID 1550057.

  21. S. Nageswararao, Existence and multiplicity for a system of fractional higher-order two-point boundary value problem, J. Appl. Math. Comput. 51 (2016), 93-107.

  22. K. R. Prasad, A. Kameswararao and S. Nageswararao, Existence of positive solutions for the system of higher order two-point boundary value problems, Proc. Indian Acad. Sci. Math. Sci. 122 (2012), No. 1, 139-152.

  23. K. R. Prasad and B. M. B. Krushna, Multiple positive solutions for a coupled system of Riemann-Liouville fractional order two-point boundary value problems, Nonlinear Stud. 20 (2013), No. 4, 501-511.

  24. I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999)

  25. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam (2006).

  26. C. Tian and Y. Liu, Multiple positive solutions for a class of fractional singular boundary value problem, Mem. Differ. Equ. Math. Phys. 56 (2012), 115-131.