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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); 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, (Contributions to the Theory of Games, Vol. I (1950), Princeton Univ. Press), 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: 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: 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); 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); 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: 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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