American Institute of Mathematical Sciences

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

Positive solutions of nonlocal fractional boundary value problems

John R. Graef 1, , Lingju Kong 1, , Qingkai Kong 2, and  Min Wang 1,

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.
Keywords: fractional calculus., positive solution, Green's function.
Mathematics Subject Classification: Primary: 34B15, 34B1.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

K. Deimling, "Nonlinear Functional Analysis'', Springer-Verlag, New York, 1985.  Google Scholar

[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.  Google Scholar

[6]

C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010), 1050-1055.  Google Scholar

[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. Google Scholar

[8]

D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones'', Academic Press, New York, 1988.  Google Scholar

[9]

R. Hilfer, "Applications of Fractional Calculus in Physics'', World Scientific, Singapore, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

V. Tarasov, "Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media'', Springer-Verlag, New York, 2011.  Google Scholar

[13]

L. Yang and H. Chen, Unique positive solutions for fractional differential equation boundary value problems, Appl. Math. Lett. 23 (2010), 1095-1098.  Google Scholar

[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.  Google Scholar

[15]

E. Zeidler, "Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems'', Springer-Verlag, New York, 1986.  Google Scholar

[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.  Google Scholar

[17]

S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl. 59 (2010), 1300-1309.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

K. Deimling, "Nonlinear Functional Analysis'', Springer-Verlag, New York, 1985.  Google Scholar

[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.  Google Scholar

[6]

C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010), 1050-1055.  Google Scholar

[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. Google Scholar

[8]

D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones'', Academic Press, New York, 1988.  Google Scholar

[9]

R. Hilfer, "Applications of Fractional Calculus in Physics'', World Scientific, Singapore, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

V. Tarasov, "Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media'', Springer-Verlag, New York, 2011.  Google Scholar

[13]

L. Yang and H. Chen, Unique positive solutions for fractional differential equation boundary value problems, Appl. Math. Lett. 23 (2010), 1095-1098.  Google Scholar

[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.  Google Scholar

[15]

E. Zeidler, "Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems'', Springer-Verlag, New York, 1986.  Google Scholar

[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.  Google Scholar

[17]

S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl. 59 (2010), 1300-1309.  Google Scholar

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