Abstract
In this paper, we employ Krasnoseľskii’s fixed point theorem to show the existence of positive solutions to three different two point fractional boundary value problems with fractional boundary conditions. Also, nonexistence results are given.
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References
Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, No 2 (2005), 495–505.
K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Equations. Springer (2010).
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.
P.W. Eloe and J.T. Neugebauer, Convolutions and Green’s functions for two families of boundary value problems for fractional differential equations. Electron. J. Differential Equations 2016, No 297 (2016), 13 p.
J.R. Graef, J. Henderson, and B. Yang, Positive solutions of a nonlinear n-th order eigenvalue problem. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13B, Suppl. Vol. (2006), 39–48.
J.R. Graef, J. Henderson, and B. Yang, Positive solutions of a non-linear higher order boundary-value problem. Electron. J. Differential Equations 2007, No 45 (2007), 10 pp.
J.R. Graef, J. Henderson, and B. Yang, Existence of positive solutions of a higher order nonlocal singular boundary value problem. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 16, Differential Equations and Dynamical Systems, Suppl. S1 (2009), 147–152.
J.R. Graef, L. Kong, M. Wang, and B. Yang, Uniqueness and parameter dependence of positive solutions of a discrete fourth-order problem. J. Difference Equ. Appl. 19, No 7 (2013), 1133–1146.
J.R. Graef, L. Kong, and B. Yang, Positive solutions of a nonlinear higher order boundary-value problem. Discrete Contin. Dyn. Syst.: Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, Suppl. (2009), 276–285.
J.R. Graef, L. Kong, and B. Yang, Positive solutions for a semipositone fractional boundary value problem with a forcing term. Fract. Calc. Appl. Anal. 15, No 1 (2012), 8–24; DOI: 10.2478/s13540-012-0002-7; https://www.degruyter.com/view/j/fca.2012.15.issue-1/issue-files/fca.2012.15.issue-1.xml.
J.R. Graef, B. Yang, Existence and nonexistence of positive solutions of a nonlinear third order boundary value problem. Proc. Colloq. Qual. Theory Differ. Equ. 8, No 9 (2008), 13 pp.
J.R. Graef and B. Yang, Positive solutions to a three point fourth order focal boundary value problem. Rocky Mountain J. Math. 44, No 3 (2014), 937–951.
J. Henderson and R. Luca, Positive solutions for system of nonlocal fractional boundary value problems. Fract. Calc. Appl. Anal. 16, No 4 (2014), 985–1008; DOI: 10.2478/s13540-013-0061-4; https://www.degruyter.com/view/j/fca.2013.16.issue-4/issue-files/fca.2013.16.issue-4.xml.
J. Ji and B. Yang, Computing the positive solutions of the discrete third-order three-point right focal boundary-value problems. Int. J. Comput. Math. 91, No 5 (2014), 996–1004.
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, No 2 (2010), 710–719.
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. 2008, No 3 (2008), 11 pp.
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Vol. 204 of North-Holland Mathematics Studies, Elsevier Sci. B.V., Amsterdam (2006).
M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations. Transl. by A.H. Armstrong; Transl. Ed. by J. Burlak, A Pergamon Press Book, The Macmillan Co., New York (1964).
J.W. Lyons and J.T. Neugebauer, Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Opuscula Math. 37, No 3 (2017), 421–434.
K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993).
I. Podlubny, Fractional Differential Equations. Vol. 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999).
M. Ur Rehman, R. Ali Khan, and P.W. Eloe, Positive solutions of nonlocal boundary value problem for higher order fractional differential system. Dynam. Systems Appl. 20, No 2-3 (2011), 169–182.
B. Yang, Upper estimate for positive solutions of the (p, n–p) conjugate boundary value problem. J. Math. Anal. Appl. 390, No 2 (2012), 535–548.
S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differential Equations 2006, No 36 (2006), 12 pp.
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Lyons, J.W., Neugebauer, J.T. Two Point Fractional Boundary Value Problems with a Fractional Boundary Condition. FCAA 21, 442–461 (2018). https://doi.org/10.1515/fca-2018-0025
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DOI: https://doi.org/10.1515/fca-2018-0025