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Positive solutions for boundary value problems of nonlinear fractional differential equations. (English) Zbl 1227.34011

The existence of at least one or two positive solutions for boundary eigenvalue problems of nonlinear fractional differential equations is established by using properties of the Green function and the Guo-Krasnosel’skii fixed point theorem on cones.

MSC:

34A08 Fractional ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

[1] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equation (1993), John Wiley: John Wiley New York · Zbl 0789.26002
[2] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[3] Podlubny, I., Fractional differential equations, (Mathematics in Science and Engineering (1999), Academic Press: Academic Press New York/London/Toronto) · Zbl 0918.34010
[4] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integral and Derivative, (Theory and Applications (1993), Gordon and Breach: Gordon and Breach Switzerland) · Zbl 0617.26004
[5] Agarwal, R. P., Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272, 368-379 (2002) · Zbl 1070.49013
[6] Agarwal, R. P.; Belmekki, Mohammed; Benchohra, Mouffak, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Advances in Difference Equations, 2009, 47 (2009), Article ID 981728 · Zbl 1182.34103
[7] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204, 609-625 (1996) · Zbl 0881.34005
[8] Zhang, S., The existence of a positive solution for nonlinear fractional differential equation, J. Math. Anal. Appl., 252, 804-812 (2000) · Zbl 0972.34004
[9] Zhang, S., Existence of positive solutions for some class of nonlinear fractional equation, J. Math. Anal. Appl., 278, 136-148 (2003) · Zbl 1026.34008
[10] Jafari, H.; Daftardar-Gejji, V., Positive solutions of nonlinear fractional boundary value problems using adomian decomposition method, Appl. Math. Comput., 180, 700-706 (2006) · Zbl 1102.65136
[11] Xu, X.; Jiang, D.; Yuan, C., Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal., 71, 4676-4688 (2009) · Zbl 1178.34006
[12] Zhang, S., Positive solutions for boundary-value problems of nonlinear fractional equations, Electron. J. Diff. Equat., 36, 1-12 (2006)
[13] Qiu, T.; Bai, Z., Existence of positive solutions for singular fractional equations, Electron. J. Diff. Equat., 146, 1-9 (2008)
[14] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional equation, J. Math. Anal. Appl., 311, 495-505 (2005) · Zbl 1079.34048
[15] Zhao, Y.; Sun, S.; Han, Z.; Li, Q., The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16, 2086-2097 (2011) · Zbl 1221.34068
[16] Zhao, Y.; Sun, S.; Han, Z.; Li, Q., Positive solutions to boundary value problems of nonlinear fractional differential equations, Abstr. Appl. Anal., 2011, 1-16 (2011)
[17] Kilbas, A. A.; Srivastava, H. H.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier Science B.V.: Elsevier Science B.V. Amsterdam · Zbl 1092.45003
[18] Krasnoselskii, M. A., Positive Solution of Operator Equation (1964), Noordhoff Groningen
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