Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions

Year 2018, Volume: 1 Issue: 1, 39 - 45, 11.03.2018

Bouteraa Noureddine , Slimane Benaicha Habib Djourdem

https://doi.org/10.32323/ujma.396363

Cited By: 5

Abstract

In this paper, we study the boundary value problem of a class of fractional differential equations involving the Riemann-Liouville fractional derivative with nonlocal integral boundary conditions. To establish the existence results for the given problems, we use the properties of the Green’s function and the monotone iteration technique, one shows the existence of positive solutions and constructs two successively iterative sequences to approximate the solutions. The results are illustrated with an example.

Keywords

Positive Solution, Iterative sequence, Green’s function

References

• [1] A. A. Kilbas, H. M. Srivastava, J. J. Trijullo, Theory and applications of fractional differential equations, Elsevier Science b. V, Amsterdam, (2006). 1.
• [2] A. Cabada, G. Wang, positive solutions of nonlinear fractional differential equations with integral boundary conditions. J. Math. Anal. Appl. 389(1), (2012), 403-411.
• [3] C. F. Li, X. N. Luo and Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59 (2010), 1363-1375.
• [4] F. Liu, K. Burrage, Novel techniques in parameter estimation for fractional dynamycal models arising from biological systems Comput. Math. Appl. 62(3), (2011), 822-833.
• [5] J. R. Graef, L. Kong, Q. Kong and M. Wang, Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions, Fract. Calc. Appl. Anal. 15 (2012), 509-528.
• [6] J. R. Graef, L. Kong, Q. Kong, Positive solution for a class of higher-order boundary value problems with fractional q-derivatives. Appl. Math. Comput. 218 (2012), 9682-9689.
• [7] J. R. Graef, L. Kong, Q. Kong, and M. Wang, Existence and uniqueness of solutions for a fractional boundary value problem with Dirichlet boundary condition, Electron. J. Qual. Theory Differ. Equ. 2013 No.55,11 pp.
• [8] M. Al-Refai, M. Hadjji, Monotone iterative sequences for nonlinear boundary value problems of fractional order. Nonlinear Anal, 74 (2011), 3531-3539.
• [9] J. Tan and C. Cheng, Fractional boundary value problems with Riemann-Liouville fractional derivatives. Bound. Value Probl. Doi: 13662-015-0413-y, (2015), 14 pages.
• [10] N. Bouteraa and S. Benaicha, Existence of solutions for three-point boundary value problem for nonlinear fractional differential equations, An. Univ. Oradea, fasc. Math. Volume 25 (2018), nr. 1. to appear.
• [11] S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, Inc.; New York, (1993). 1, 1.3, 1
• [12] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993).
• [13] S. liu, H. Li and Q. Dai, Nonlinear fractional differential equations with nonlocal integral boundary conditions, Bound.Value. Prob. (2015), 11 pages.
• [14] V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 21 (8) (2008), 828-834.
• [15] X. L. Han, H. L. Gao, Existence of positive solutions for eigenvalue problem of nonlinear fractional differential equations. Adv. Differ. Equ. 2012 66 (2012).
• [16] Y. Sun, M. Zhao, Positive solutions for a class of fractional differential equations with integral boundary conditions. Appl. Math. Lett, 34 (2014), 17-21.
• [17] Y. F. Xu, Fractional boundary value problems with integral and anti-periodic boundary conditions. Bull. Malays. Math. Sci. Soc. 39, (2016), 571-587.
• [18] Y. Qiao and Z. zhou, Existence of solutions for a class of fractional differential equations with integral and anti-periodic boundary conditions, Bound. Value Probl. Doi: 13661-016-0547-x, (2017), 9 pages.
• [19] Z. B. Bai, W. C. Sun and W. Zhang, Positive solutions for boundary value problems of singular fractional differential equations. Abstr. Appl. Anal. 2013, Article ID 129640 (2013).