Chai, Guoqing Existence results for boundary value problems of nonlinear fractional differential equations. (English) Zbl 1231.34007 Comput. Math. Appl. 62, No. 5, 2374-2382 (2011). Summary: We consider the existence of solutions for the nonlinear fractional differential equation with the boundary value conditions where and are the standard Caputo derivative with \(1<\alpha \leq 2\), \(r\neq 0\). By using the contraction mapping principle and the Schauder fixed point theorem, some existence results are obtained. In addition, Lemma 2.6 in this paper is a valuable tool in seeking solvability of the fractional differential equations. Cited in 15 Documents MSC: 34A08 Fractional ordinary differential equations 45J05 Integro-ordinary differential equations Keywords:fractional differential equations; boundary value problem; existence of solution; fixed point theorem PDFBibTeX XMLCite \textit{G. Chai}, Comput. Math. Appl. 62, No. 5, 2374--2382 (2011; Zbl 1231.34007) Full Text: DOI References: [1] Glockle, W. G.; Nonnenmacher, T. F., A fractional calculus approach of self-similar protein dynamics, Biophys. J., 68, 46-53 (1995) [2] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore · Zbl 0998.26002 [3] Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F., Relaxation in filled polymers: a fractional calculus approach, J. Chem. Phys., 103, 7180-7186 (1995) [4] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 [5] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5, 367-386 (2002) · Zbl 1042.26003 [6] Kilbas, A. A.; Srivastava, Hari M.; Trujillo, Juan J., (North-Holland Mathematics Studies. North-Holland Mathematics Studies, Theory and Applications of Fractional Differential Equations, vol. 204 (2006), Elsevier Science: Elsevier Science B.V, Amsterdam) [7] Lakshmikantham, V.; Leela, S.; Vasundhara, J., Theory of Fractional Dynamic Systems (2009), Cambridge Academic Publishers: Cambridge Academic Publishers Cambridge · Zbl 1188.37002 [8] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivaes Theory and Applications (1993), Gordon and Breach Science Publisher · Zbl 0818.26003 [9] Agarwal, R. P.; Benchohra, M.; Hamani, S., A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109, 973-1033 (2010) · Zbl 1198.26004 [10] Lakshmikanthama, V.; Vatsala, A. S., General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett., 21, 828-834 (2008) · Zbl 1161.34031 [11] Lakshmikanthama, V., Theory of fractional functional differential equations, Nonlinear Anal., 69, 3337-3343 (2008) · Zbl 1162.34344 [12] Agarwal, Ravi P.; O’Regan, Donal; Stanek, Svatoslav, Positive solutions for Dirichlet problems of singular nonlinear fractional, differential equations, J. Math. Anal. Appl., 371, 57-68 (2010) · Zbl 1206.34009 [13] McRae, F. A., Monotone iterative technique and existence results for fractional differential equtions, Nonlinear Anal., 71, 6093-6096 (2009) · Zbl 1260.34014 [14] Benchohra, M.; Hamani, S.; Ntouyas, S. K., Boundary value problems for differential equational with fractional order and nonlocal condtions, Nonlinear Anal., 71, 2391-2396 (2009) · Zbl 1198.26007 [15] Agarwal, R. P.; Zhou, Yong; He, Yunyun, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59, 1095-1100 (2010) · Zbl 1189.34152 [16] Ahmad, Bashir; Nieto, Juan J., Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 58, 1838-1843 (2009) · Zbl 1205.34003 [17] Shuqin Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differential Equations, 2006, (36), 1-12.; Shuqin Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differential Equations, 2006, (36), 1-12. · Zbl 1096.34016 [18] Rehman, Mujeeb ur; Khan, Rahmat Ali, Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett., 1038-1044 (2010) · Zbl 1214.34007 [19] Balachandran, Krishnan; Trujillo, Juan J., The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Anal., 72, 4587-4593 (2010) · Zbl 1196.34007 [20] Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A., Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338, 2, 1340-1350 (2008) · Zbl 1209.34096 [21] Baleanu, Dumitru; Mustafa, Octavian G., On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl., 59, 1835-1841 (2010) · Zbl 1189.34006 [22] El-Shahed, Moustafa, Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order, Comput. Math. Appl., 59, 3438-3443 (2010) · Zbl 1197.34003 This reference list is based on information provided by the publisher or from digital mathematics libraries. 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