The multiplicity of positive solutions for systems of fractional boundary value problems

Abstract

This paper focuses on the multiple positive solutions for a coupled system of nonlinear boundary value problems of fractional order. Our approach is based on a fixed point theorem due to Bai and Ge. Also, an example is given to demonstrate the applicability of our main result.

Keywords

• multiple positive solution,fractional differential equation,fixed point theorem

References

1. [1] Z.E. Abidine, Multiple Positive Solutions for a Coupled System of Nonlinear Fractional Differential Equations on the Half-line, Mediterr. J. Math. 14, Article No: 142, 16 pages, 2017.
2. [2] B. Ahmad and J.J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58, 1838-1843, 2009.
3. [3] B. Ahmad, J.J. Nieto, A. Alsaedi and M.H. Aqlan, A Coupled System of Caputo-Type Sequential Fractional Differential Equations with Coupled (Periodic/Anti-periodic Type) Boundary Conditions, Mediterr. J. Math. 14, Article No: 227, 2017.
4. [4] Z. Bai and W. Ge, Existence of three positive solutions for some second-order boundary value problems, Comput. Math. Appl. 48, 699-707, 2004.
5. [5] T.S. Cerdik, N.A. Hamal and F. Yoruk Deren, Existence of solutions for nonlinear fractional differential equations with m-point integral boundary conditions, Dynam. Systems Appl. 24, 283-294, 2015.
6. [6] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
7. [7] J. Henderson and R. Luca, Positive solutions for a system of nonlocal fractional boundary value problems, Fract. Calc. Appl. Anal. 16 (4), 985-1008, 2013.
8. [8] J. Henderson and R. Luca, Positive solutions for a system of semipositone coupled fractional boundary value problems, Bound. Value Probl. 2016, Article No: 61, 2016.

9. [9] J. Henderson and R. Luca, Systems of Riemann-Liouville fractional equations with multi-point boundary conditions, Appl. Math. Comput. 309, 303-323, 2017.
10. [10] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, in: North-Holland Mathematics Studies 204, Elsevier Science B.V, Amsterdam, 2006.
11. [11] Y. Liu, New existence results for positive solutions of boundary value problems for coupled systems of multi-term fractional differential equations, Hacet. J. Math. Stat. 45 (2), 391-416, 2016.
12. [12] N. Nyamoradi, Multiple positive solutions for fractional differential systems, Ann Univ Ferrara 58, 359-369, 2012.
13. [13] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
14. [14] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22, 64-69, 2009.
15. [15] Y.Wang, Positive solutions for a system of fractional integral boundary value problem, Bound. Value Probl. 2013, Article No: 256, 2013.
16. [16] A. Yang and W. Ge, Positive Solutions for Boundary Value Problems of N-Dimension Nonlinear Fractional Differential System, Bound. Value Probl. 2008, Article ID 437453, 15 pages, 2008.
17. [17] A. Yang and H. Wang, Positive solutions for higher-order nonlinear fractional differential equation with integral boundary condition, Electron. J. Qual. Theory Differ. Equ. 2011 (1), 1-15, 2011.