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Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. (English) Zbl 1189.34014

Summary: We are concerned with the nonlinear differential equation of fractional order \[ D^{\alpha}_{0+}u(t)+f(t,u(t))=0,~0<t<1,~1<\alpha\leq 2, \] where \(D^{\alpha}_{0+}\) is the standard Riemann-Liouville fractional derivative, subject to the boundary conditions \(u(0)=0\), \(D^{\beta}_{0+}u(1)=aD^{\beta}_{0+}u(\xi)\). We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems.

MSC:

34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations
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[1] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., (Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier: Elsevier Amsterdam) · Zbl 1092.45003
[2] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[3] Podlubny, I., Fractional Differential Equations, Mathematics in Science and Engineering (1999), Academic Press: Academic Press New York
[4] Bakakhani, A.; Gejji, V. D., Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl., 278, 434-442 (2003) · Zbl 1027.34003
[5] Delbosco, D.; Rodina, L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204, 609-625 (1996) · Zbl 0881.34005
[6] El-Sayed, A. M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal., 33, 181-186 (1998) · Zbl 0934.34055
[7] Lakshmikantham, V., Theory of fractional functional differential equations, Nonlinear Anal., 69, 3337-3343 (2008) · Zbl 1162.34344
[8] Zhang, S. Q., Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl., 278, 136-148 (2003) · Zbl 1026.34008
[9] Zhou, Yong, Existence and uniqueness of solutions for a system of fractional differential equations, J. Frac. Calc. Appl. Anal., 12, 195-204 (2009) · Zbl 1396.34003
[10] Zhou, Yong, Existence and uniqueness of fractional functional differential equations with unbounded delay, Int. J. Dyn. Syst. Differ. Equ., 1, 4, 239-244 (2008) · Zbl 1175.34081
[11] Bai, Z. B.; Lü, H. S., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505 (2005) · Zbl 1079.34048
[12] Kosmatov, N., A singular boundary value problem for nonlinear differential equations of fractional order, J. Appl. Math. Comput. (2008)
[13] Kaufmann, E. R.; Mboumi, E., Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Diff. Equ., 2008, 3, 1-11 (2008) · Zbl 1183.34007
[14] Bai, C. Z., Triple positive solutions for a boundary value problem of nonlinear fractional differential equation, Electron. J. Qual. Theory Diff. Equ., 2008, 24, 1-10 (2008)
[15] Krasnosel’skii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations (1964), Pergamon: Pergamon Elmsford, (A.H. Armstrong, Trans.) · Zbl 0111.30303
[16] Agarwal, R. P.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0960.54027
[17] Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach space, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033
[18] Granas, A.; Guenther, R. B.; Lee, J. W., Some general existence principle in the Caratheodory theory of nonlinear systems, J. Math. Pures Appl., 70, 153-196 (1991) · Zbl 0687.34009
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