**AUTHORS:**Qiuyan Zhong, Xingqiu Zhang

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**ABSTRACT:**
In this paper, we investigate the existence of positive solutions for higher-order fractional differential
equations with p-Laplacian operator and nonlocal boundary conditions. By means of the properties of the corresponding
Green function together with monotone iterative technique, we obtain not only the existence of positive
solutions for the problems, but also establish iterative schemes for approximating the solutions. The nonlinearity
permits singularities at t = 0 and/or t = 1.

**KEYWORDS:**
Fractional differential equations; Integral boundary value problem; p-Laplacian; Positive solution;
Successive iteration

**REFERENCES:**

[1] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integral and Derivative, in: Theory and Applications, Gordon and Breach, Switzerland, 1993.

[2] I. Podlubny, Fractional Differential Equations, in: Mathematics in science and Engineering, vol. 198, Academic Press, New York, London, Toronto, 1999.

[3] A. A. Kilbas, H. M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[4] Y. Wang, L. Liu, Y. Wu, Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal. 74, 2011, pp. 3599-3605.

[5] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389, 2012, pp. 403-411.

[6] M. Feng, X. Zhang, W. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Boundary Value Problems, Volume 2011, 720702 (2011)

[7] L. Wang, X. Zhang, Existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter, J. Appl. Math. Comput. 44, 2014, pp. 293-316.

[8] Y. Sun, M. Zhao, Positive solutions for a class of fractional differential equations with integral boundary conditions, Appl. Math. Lett. 34, 2014, pp. 17-21.

[9] X. Zhang, Y. Han, Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett. 25, 2012, pp. 555-560.

[10] X. Zhang, Positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions, Boundary Value Problems 2012, 123 (2012)

[11] X. Zhang, Nontrivial solutions for a class of fractional differential equations with integral boundary conditions and a parameter in a Banach Space with Lattice, Abstr. Appl. Anal. 2012, 391609, (2012)

[12] X. Zhang, L. Wang, Q. Sun, Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter, Appl. Math. Comput. 226, 2014, pp. 708-718.

[13] X. Zhang, L. Liu, Y. Wu, Y. Lu, The iterative solutions of nonlinear fractional differential equations, Appl. Math. Comput. 219, 2013, pp. 4680-4691.

[14] S. Li, X. Zhang, Y. Wu, L. Caccetta, Extremal solutions for p-Laplacian differential systems via iterative computation, Appl. Math. Lett. 26, 2013, pp. 1151-1158.

[15] Y. Tian, X. Li, Existence of positive solution to boundary value problem of fractional differential equations with p-Laplacian operator, J. Appl. Math. Comput., 47, 2015, pp. 237-248.

[16] X. Zhang, Positive solutions for singular higherorder fractional differential equations with nonlocal conditions, J. Appl. Math. Comput., 49, 2015, pp. 69-89.