Lower and Upper Solutions for General Two-Point Fractional Order Boundary Value Problems

Abstract

This paper establishes the existence of a positive solution of fractional order two-point boundary value problem, D q1 a+ y t + f t, y t = 0, t ∈ [a, b], y a = 0, y ′ a = 0, αDq2 a+ y b − βDq3 a+ y a = 0, where D qi a+ , i = 1, 2, 3 are the standard Riemann-Liouville fractional order derivatives, 2 < q1 ≤ 3, 0 < q2, q3 < q1, α, β are positive real numbers and b > a ≥ 0, by an application of lower and upper solution method and fixed-point theorems

Keywords

• Fractional derivative,Boundary value problem,Two-point,Green’s function,Positive solution

References

1. Bai, Z. and L¨u, H., (2005), Positive solutions for boundary value problems of nonlinear fractional differential equations, J. Math. Anal. Appl., 311, pp. 495–505.
2. Benchohra, M., Henderson, J., Ntoyuas, S. K. and Ouahab, A., (2008), Existence results for fractional order functional differential equations with inŞnite delay, J. Math. Anal. Appl., 338, pp. 1340-1350. [3] Guo, D. and Zhang, J., (1985), Nonlinear Fractional Analysis, Science and Technology Press, Jinan, China.
3. Habets, P. and Zanolin, F., (1994), Upper and lower solutions for a generalized Emden-Fowler equa- tion, J. Math. Anal. Appl., 181, no. 3, pp. 684-700.
4. Kauffman, E. R. and Mboumi, E., (2008), Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ., 3, pp. 1-11.
5. Khan, R. A., Rehman M. and Henderson, J., (2011), Existence and uniqueness of solutions for nonlin- ear fractional differential equations with integral boundary conditions, Fractional Differential Calculus, 1, pp. 29–43.
6. Kilbas, A. A., Srivasthava, H. M. and Trujillo, J. J., (2006), Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam.
7. Lee, Y. H., (1997), A multiplicity result of positive solutions for the generalized Gelfand type singular boundary value problems, Proceedings of the Second World Congress of Nonlinear Analysis, Part 6 (Athens, 1996); Nonlinear Anal., 30, no. 6, pp. 3829-3835.
8. Li, F., Sun, J. and Jia, M., (2011), Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order, Appl. Math. Comput., 217, no. 9, pp. 4840-4847.

9. Liang, S. and Zhang, J., (2006), Positive solutions for boundary value problems of nonlinear fractional differential equations, Elec. J. Diff. Eqns., 36, pp. 1-12.
10. Podulbny, I., (1999), Fractional Differential Equations, Academic Press, San Diego.
11. Prasad, K. R. and Krushna, B. M. B., (2013), Multiple positive solutions for a coupled system of Riemann-Liouville fractional order two-point boundary value problems, Nonlinear Stud., vol. 20, no.4, pp. 501-511.
12. Prasad, K. R. and Krushna, B. M. B., (2014), Eigenvalues for iterative systems of Sturm-Liouville fractional order two-point boundary value problems, Fract. Calc. Appl. Anal., vol. 17, no. 3, pp. 638-653, DOI: 10.2478/s13540-014-0190-4.
13. Shi, A. and Zhang, S., (2009), Upper and lower solutions method and a fractional differential equation boundary value problem, Electron. J. Qual. Theory Differ. Equ., no. 30, pp. 1-13.
14. Su, X. and Zhang, S., (2009), Solutions to boundary value problems for nonlinear differential equations of fractional order, Elec. J. Diff. Eqns., 26, pp. 1-15.
15. Zhang, S., (2006), Existence of solutions for a boundary value problem of fractional order, Acta Math. Sci., 26B, pp. 220-228.