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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.  Google Scholar

[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.  Google Scholar

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

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

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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. Google Scholar

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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. Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[12]

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

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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.  Google Scholar

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

[15]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[15]

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

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