# American Institute of Mathematical Sciences

2013, 2013(special): 283-290. doi: 10.3934/proc.2013.2013.283

## Positive solutions of nonlocal fractional boundary value problems

 1 Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States, United States 2 Department of Mathematics, Northern Illinois University, DeKalb, Il 60115

Received  August 2012 Published  November 2013

The authors study a type of nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed. Then a criterion for the existence of at least one positive solution is obtained by using fixed point theory on cones.
Citation: John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283
##### References:
 [1] R. Agarwal, D. O'Regan, and S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl. 371 (2010), 57-68. [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 (2009), 1838-1843. [3] Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), 495-505. [4] K. Deimling, "Nonlinear Functional Analysis'', Springer-Verlag, New York, 1985. [5] M. Feng, X. Zhang, and W. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl. (2011), Art. ID 720702, 20 pp. [6] C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010), 1050-1055. [7] J. R. Graef, L. Kong, Q. Kong, and M. Wang, Fractional boundary value problems with integral boundary conditions, Appl. Anal. 92 (2013), 2008-2020. [8] D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones'', Academic Press, New York, 1988. [9] R. Hilfer, "Applications of Fractional Calculus in Physics'', World Scientific, Singapore, 2000. [10] D. Jiang and C. Yuan, The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application, Nonlinear Anal. 72 (2010), 710-719. [11] Q. Kong and M. Wang, Positive solutions of nonlinear fractional boundary value problems with Dirichlet boundary conditions, Electron. J. Qual. Theory Differ. Equ., No. 17 (2012), 1-13. [12] V. Tarasov, "Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media'', Springer-Verlag, New York, 2011. [13] L. Yang and H. Chen, Unique positive solutions for fractional differential equation boundary value problems, Appl. Math. Lett. 23 (2010), 1095-1098. [14] W. Yang, Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions, Comput. Math. Appl. 63 (2012), 288-297. [15] E. Zeidler, "Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems'', Springer-Verlag, New York, 1986. [16] C. Zhai and M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems, Nonlinear Anal., 75 (2012), 2542-2551. [17] S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl. 59 (2010), 1300-1309.

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##### References:
 [1] R. Agarwal, D. O'Regan, and S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl. 371 (2010), 57-68. [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 (2009), 1838-1843. [3] Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), 495-505. [4] K. Deimling, "Nonlinear Functional Analysis'', Springer-Verlag, New York, 1985. [5] M. Feng, X. Zhang, and W. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl. (2011), Art. ID 720702, 20 pp. [6] C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010), 1050-1055. [7] J. R. Graef, L. Kong, Q. Kong, and M. Wang, Fractional boundary value problems with integral boundary conditions, Appl. Anal. 92 (2013), 2008-2020. [8] D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones'', Academic Press, New York, 1988. [9] R. Hilfer, "Applications of Fractional Calculus in Physics'', World Scientific, Singapore, 2000. [10] D. Jiang and C. Yuan, The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application, Nonlinear Anal. 72 (2010), 710-719. [11] Q. Kong and M. Wang, Positive solutions of nonlinear fractional boundary value problems with Dirichlet boundary conditions, Electron. J. Qual. Theory Differ. Equ., No. 17 (2012), 1-13. [12] V. Tarasov, "Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media'', Springer-Verlag, New York, 2011. [13] L. Yang and H. Chen, Unique positive solutions for fractional differential equation boundary value problems, Appl. Math. Lett. 23 (2010), 1095-1098. [14] W. Yang, Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions, Comput. Math. Appl. 63 (2012), 288-297. [15] E. Zeidler, "Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems'', Springer-Verlag, New York, 1986. [16] C. Zhai and M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems, Nonlinear Anal., 75 (2012), 2542-2551. [17] S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl. 59 (2010), 1300-1309.
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