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Boundary value problems for differential equations with fractional order and nonlocal conditions. (English) Zbl 1198.26007
The authors establish sufficient conditions for a class of non-local conditions boundary value problems for fractional differential equation of order between 1 and 2 and involving the Caputo derivative.

MSC:
26A33 Fractional derivatives and integrals
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Diethelm, K.; Freed, A.D., On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, (), 217-224
[2] Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. syst. signal process., 5, 81-88, (1991)
[3] Glockle, W.G.; Nonnenmacher, T.F., A fractional calculus approach of self-similar protein dynamics, Biophys. J., 68, 46-53, (1995)
[4] Hilfer, R., Applications of fractional calculus in physics, (2000), World Scientific Singapore · Zbl 0998.26002
[5] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[6] 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)
[7] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York, London · Zbl 0428.26004
[8] El-Sayed, A.M.A., Fractional order evolution equations, J. fract. calc., 7, 89-100, (1995) · Zbl 0839.34069
[9] El-Sayed, A.M.A., Fractional order diffusion-wave equations, Internat. J. theoret. phys., 35, 311-322, (1996) · Zbl 0846.35001
[10] El-Sayed, A.M.A.; Ibrahim, A.G., Multivalued fractional differential equations, Appl. math. comput., 68, 15-25, (1995) · Zbl 0830.34012
[11] Kilbas, A.A.; Srivastava, Hari M.; Juan Trujillo, J., ()
[12] Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002
[13] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[14] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives. theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[15] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005
[16] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[17] Diethelm, K.; Walz, G., Numerical solution of fractional order differential equations by extrapolation, Numer. algorithms, 16, 231-253, (1997) · Zbl 0926.65070
[18] El-Sayed, A.M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear anal., 33, 181-186, (1998) · Zbl 0934.34055
[19] Kilbas, A.A.; Marzan, S.A., Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differ. equ., 41, 84-89, (2005) · Zbl 1160.34301
[20] Momani, S.M.; Hadid, S.B., Some comparison results for integro-fractional differential inequalities, J. fract. calc., 24, 37-44, (2003) · Zbl 1057.45003
[21] Momani, S.M.; Hadid, S.B.; Alawenh, Z.M., Some analytical properties of solutions of differential equations of noninteger order, Int. J. math. math. sci., 2004, 697-701, (2004) · Zbl 1069.34002
[22] Podlubny, I.; Petraš, I.; Vinagre, B.M.; O’Leary, P.; Dorčak, L., Analogue realizations of fractional-order controllers. fractional order calculus and its applications, Nonlinear dynam., 29, 281-296, (2002) · Zbl 1041.93022
[23] Yu, C.; Gao, G., Existence of fractional differential equations, J. math. anal. appl., 310, 26-29, (2005) · Zbl 1088.34501
[24] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal. TMA, 69, 8, 2677-2682, (2008) · Zbl 1161.34001
[25] Lakshmikantham, V.; Vatsala, A.S., Theory of fractional differential inequalities and applications, Commun. appl. anal., 11, 3-4, 395-402, (2007) · Zbl 1159.34006
[26] Lakshmikantham, V.; Vatsala, A.S., General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. lett., 21, 8, 828-834, (2008) · Zbl 1161.34031
[27] Byszewski, L.; Lakshmikantham, V., Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. anal., 40, 11-19, (1991) · Zbl 0694.34001
[28] Byszewski, L., Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. math. anal. appl., 162, 494-505, (1991) · Zbl 0748.34040
[29] L. Byszewski, Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal Cauchy problem, Selected problems of mathematics, 50th Anniv. Cracow Univ. Technol. Anniv., Issue 6, Cracow Univ. Technol., Krakow, 1995, pp. 25-33
[30] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. calc. appl. anal., 5, 367-386, (2002) · Zbl 1042.26003
[31] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differential Equations 2006 (36) 1-12
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