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Some new existence results for fractional differential inclusions with boundary conditions. (English) Zbl 1165.34313
Summary: This paper is mainly concerned with the existence of solutions for a certain class of fractional differential inclusions with boundary conditions. By using Bohnenblust-Karlin’s fixed point theorem, a main existence theorem is obtained. As an application of this main theorem, we establish two existence results when the multi-valued nonlinearity \(F\) has sub-linear or linear growth in the state variable \(y\). Our results are even new when applied to a corresponding single-valued problem.

MSC:
34A60 Ordinary differential inclusions
26A33 Fractional derivatives and integrals
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Abbasbandy, S.; Nieto, J.J.; Alavi, M., Tuning of reachable set in one dimensional fuzzy differential inclusions, Chaos solitons & fractals, 26, 1337-1341, (2005) · Zbl 1073.65054
[2] Benchohra, M.; Henderson, J.; Ntouyas, S.K., On nonresonance second order impulsive functional differential inclusions with nonlinear boundary conditions, Can. appl. math. Q., 14, 21-32, (2006) · Zbl 1146.34055
[3] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. (in press) · Zbl 1209.34096
[4] Bohnenblust, H.F.; Karlin, S., On a theorem of ville, (), 155-160 · Zbl 0041.25701
[5] Chang, Y.K.; Li, W.T., Existence results for second order impulsive functional differential inclusions, J. math. anal. appl., 301, 477-490, (2005) · Zbl 1067.34083
[6] Chang, Y.K.; Li, W.T.; Nieto, J.J., Controllability of evolution differential inclusions in Banach spaces, Nonlinear anal. TMA, 67, 623-632, (2007) · Zbl 1128.93005
[7] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equations, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005
[8] Deimling, K., Multivalued differential equations, (1992), De Gruyter Berlin · Zbl 0760.34002
[9] Frigon, M., Systems of first order differential inclusions with maximal monotone terms, Nonlinear anal. TMA, 66, 2064-2077, (2007) · Zbl 1121.34017
[10] EI-Sayed, A.M.A., On the fractional differential equations, Appl. math. comput., 49, 205-213, (1992) · Zbl 0757.34005
[11] EI-Sayed, A.M.A, Nonlinear functional differential equations of arbitrary order, Nonlinear anal. TMA, 33, 181-186, (1998) · Zbl 0934.34055
[12] EI-Sayed, A.M.A; Ibrahim, A.-G., Set-valued integral equations of fractional-orders, Appl. math. comput., 118, 113-121, (2001) · Zbl 1024.45003
[13] Hu, S.; Papageorgiou, N., Handbook of multivalued analysis, (1997), Kluwer Dordrecht, Boston · Zbl 0887.47001
[14] Ibrahin, R.W.; Momani, S., On the existence and uniqueness of solutions of a class of fractional differential equations, J. math. anal. appl., 334, 1-10, (2007)
[15] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal. TMA (in press) · Zbl 1162.34344
[16] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA (in press) · Zbl 1161.34001
[17] Lasota, A.; Opial, Z., An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. acad. Pol. sci., ser.sci. math. astronom. phys., 13, 781-786, (1965) · Zbl 0151.10703
[18] Smirnov, G.V., Introduction to the theory of differential inclusions, (2002), American Mathematical Society Providence, RI · Zbl 0992.34001
[19] Sun, J.P.; Zhao, Y.H., Multiplicity of positive solutions of a class of nonlinear fractional differential equations, Comput. math. appl., 49, 73-80, (2005) · Zbl 1085.34501
[20] Yu, C.; Gao, G., Existence of fractional differential equations, J. math. anal. appl., 310, 26-29, (2005) · Zbl 1088.34501
[21] Yu, C.; Gao, G., On the solution of nonlinear fractional order differential equation, Nonlinear anal. TMA, 63, e971-e976, (2005) · Zbl 1224.34005
[22] Zhang, S., The existence of a positive solution for a nonlinear fractional differential equation, J. math. anal. appl., 252, 804-812, (2000) · Zbl 0972.34004
[23] Zhang, S., Existence of positive solution for some class of nonlinear fractional differential equations, J. math. anal. appl., 278, 136-148, (2003) · Zbl 1026.34008
[24] Zhang, S., Existence of positive solutions for fractional differential equations with riemann – liouville left-sided and right-hand fractional derivatives, Electron. J. differential equations2004, 1-12, (2004) · Zbl 1050.34003
[25] Zhang, S., Existences of solutions for a boundary value problem of fractional order, Acta math. sci., 26 B, 220-228, (2006)
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