Existstence and uniqueness of positive solution for a class of boundary value problem including fractional differential equation

Author
asghar Ahmadkhanlu
Shahid Madani University
10.29252/mmr.3.1.1
Abstract
In this paper we investigate a kind of boundary value problem involving a fractional differential equation. We study the existence of positive solutions of the problem that fractional derivative is the Reimann-Liouville fractional derivative. At first the green function is computed then it is proved that the green function is positive. We present necessary and sufficient conditions for existence of positive solution by calculating supremum of integral of green function over the solution interval and by use of some expansions of contraction mapping that are presented recently. For this purpose, at first the existence and uniqueness of solution for the problem, by use of existence of lower solution for the problem and expansion of contraction mapping on ordered space, is proved. Then by use of another expansion of contraction mapping on ordered spaces, the existence and uniqueness of positive solution is proved. Also, an example is presented to illustrate the proven results.
Keywords
Boundary value problem
Fractional differential equations
Reimman Liouville fractional derivatve
Fixed Point theorem
1. Agarwal, Ravi P., "Formulation of Euler-Lagrange equations for fractional variational problems," J. Math. Anal. Appl )2002) 368-379. 2. Harandi, Amini A., Emami H., "A fixed point theorem for contraction type maps in partially orderd metric spaces and application to ordinary differential equations," Non. Anal., 72 (2010) 2238-2242. 3. Miller, Kenneth S., Ross, Bertram, "An Introduction to the Fractional Calculus and Fractional Differential Equation", Wiley ,NewYork (1993). 4. Oldham, Keith B., Spanier, Jerome, " The Fractional Calculus: Integrations and Differentiations of Arbitrary Order", Academic Press,New York (1974). 5. Samko, Stefan G., Kilbas, Anatoly A., Marichev, Oleg I., "Fractional Integral and Derivative: Theory and Applications", Gordon & Breach,Yverdon (1993). 6. Delbosco D., Rodino L., "Existence and uniqueness for a nonlinear fractional differential equations", J. Math. Anal. Appl. 204 (1996) 609-625. 7. Zhang S., "The existence of a positive solution for nonlinear fractional differential equation", J. Math. Anal. Appl., 252 (2000) 804-812. 8. Jafari H., Daftardar V. Gejji, "Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method," Appl. Math. Comput., 180 (2006) 700-706. 9. Bai, Zhanbing; Lü, Haishen, "Positive solutions for a boundary value problem of nonlinear fractional differential equation", J. Math. Anal. and App., 311 (2005) 495-505. 10. Kaufmann E. R., Emboumi E., "The existence of a positive solution for a nonlinear fractional differential equation", J. Qual. Theory Differ. Equ., 3 (2008) 1-11. 11. Liu, Y., "Positive solutions for Singular FDES", U. P. B. Sci. Bull., Series A, 73 (2011) 89-100. 12. Kilbas A. A., Srivastava H. H., Trujillo J. J., "Theory and Applications of Fractional Differential Equations", Elsevier, Amsterdam (2006). 13. Podlubny, Igor, "Fractional Differential Equations", Academic Press,New York (1999). 14. Karapinar E., "α-ψ-Geraghty contraction type mappings and some related fixed point results," Filomat, 28 (1) (2014) 37-48. 15. Geraghty M., "On contraction mappings", Pros. Amer. Math. Soc. ,40 (1973) 604-608. 16. Harjani J., Sadarangani K., "Fixed point theorems for weakly contractive mappings in partially orderd sets", Nonlinear Analysis:Theory method and applications, 71 (7) (2009) 3403-3410. 17. Nieto J. J., Lopez R., Rodriguez, "Contractive mapping theorems in partially orderd sets and application to ordinary differential equations", Order, 22 (3)(2005) 223-239