American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 615-620. doi: 10.3934/proc.2015.0615

Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition

Christina A. Hollon 1, and  Jeffrey T. Neugebauer 1,

1. 

Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, Kentucky 40475, United States, United States

Received  September 2014 Revised  May 2015 Published  November 2015

In this paper, we apply Krasnosel'skii's cone expansion and compression fixed point theorem to show the existence of at least one positive solution to the nonlinear fractional boundary value problem $D^\alpha_{0^+} u + a(t)f(u)=0$, $0 < t < 1$, $1 < \alpha \le 2$, satisfying boundary conditions $u(0)=D^\beta_{0^+} u(1)=0$, $0\le\beta\le1$.
Keywords: positive solution., Fractional derivative.
Mathematics Subject Classification: Primary: 34B18, 26A33; Secondary: 34B1.
Citation: Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615
References:
[1]

B. Ahmad and J. J. Nieto, Riemann-Liouville fractional differential equations with fractional boundary conditions, Fixed Point Theory, 13 (2012), 329-336.

[2]

Z. Bai and H. Lu, Positive solutions for boundary value problems of nonlinear fractional differential equations, J. Math. Anal. Appl., 311 (2005), 495-505.

[3]

V. Daftardar-Genjji, Positive solutions of a system of non-autonomous nonlinear fractional differential equations, J. Math. Anal. Appl., 302 (2004), 56-64.

[4]

K. Diethelm, The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004, Springer-Verlag, Berlin, 2010.

[5]

P. W. Eloe and J. T. Neugebauer, Conjugate points for fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 855-871.

[6]

P. W. Eloe, J. W. Lyons, and J. T. Neugebauer, An ordering on Green's functions for a family of two-point boundary value problems for fractional differential equations, Commun. Appl. Anal., 19 (2015), 453-462.

[7]

J. R. Graef and X. Liu, Existence of positive solutions of fractional boundary value problems involving bounded linear operators, J. Nonlinear Funct. Anal., 2014 (2014), 1-23.

[8]

E. R. Kaufmann and E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ., 17 (2014), 855-871.

[9]

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Math. Stud., vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[10]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, (English), Translated by A. H. Armstrong, A Pergamon Press Book, MacMillan, New York, 1964.

[11]

R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28 (1979), 673-688.

[12]

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.

[13]

G. Wang, S. K. Ntouyas, and L. Zhang, Positive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument, Adv. Difference Equ., 2011 (2011), Article ID 2.

[14]

S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 1 (2013), 12-22.

[15]

S. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Electron. J. Diff. Eqns., 2006 (2006), 1-12.

show all references

References:
[1]

B. Ahmad and J. J. Nieto, Riemann-Liouville fractional differential equations with fractional boundary conditions, Fixed Point Theory, 13 (2012), 329-336.

[2]

Z. Bai and H. Lu, Positive solutions for boundary value problems of nonlinear fractional differential equations, J. Math. Anal. Appl., 311 (2005), 495-505.

[3]

V. Daftardar-Genjji, Positive solutions of a system of non-autonomous nonlinear fractional differential equations, J. Math. Anal. Appl., 302 (2004), 56-64.

[4]

K. Diethelm, The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004, Springer-Verlag, Berlin, 2010.

[5]

P. W. Eloe and J. T. Neugebauer, Conjugate points for fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 855-871.

[6]

P. W. Eloe, J. W. Lyons, and J. T. Neugebauer, An ordering on Green's functions for a family of two-point boundary value problems for fractional differential equations, Commun. Appl. Anal., 19 (2015), 453-462.

[7]

J. R. Graef and X. Liu, Existence of positive solutions of fractional boundary value problems involving bounded linear operators, J. Nonlinear Funct. Anal., 2014 (2014), 1-23.

[8]

E. R. Kaufmann and E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ., 17 (2014), 855-871.

[9]

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Math. Stud., vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[10]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, (English), Translated by A. H. Armstrong, A Pergamon Press Book, MacMillan, New York, 1964.

[11]

R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28 (1979), 673-688.

[12]

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.

[13]

G. Wang, S. K. Ntouyas, and L. Zhang, Positive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument, Adv. Difference Equ., 2011 (2011), Article ID 2.

[14]

S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 1 (2013), 12-22.

[15]

S. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Electron. J. Diff. Eqns., 2006 (2006), 1-12.

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