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The method of upper and lower solutions and impulsive fractional differential inclusions. (English) Zbl 1221.49060
Summary: The concept of lower and upper solutions combined with the fixed point theorem of Bohnenblust-Karlin is used to investigate the existence of solutions for a class of the initial value problem for impulsive differential inclusions involving the Caputo fractional derivative.

MSC:
49N25Impulsive optimal control problems
49J27Optimal control problems in abstract spaces (existence)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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References:
[1] Diethelm, K.; Freed, A. D.: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. Scientifice computing in chemical engineering II-computational fluid dynamics, reaction engineering and molecular properties, 217-224 (1999)
[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) · Zbl 0998.26002
[5] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and fractional calculus in continuum mechanics, 291-348 (1997) · 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) · Zbl 0292.26011
[8] Lakshmikantham, V.; Leela, S.; Vasundhara, J.: Theory of fractional dynamic systems. (2009) · Zbl 1188.37002
[9] Kilbas, A. A.; Srivastava, Hari M.; Trujillo, Juan J.: Theory and applications of fractional differential equations. North-holland mathematics studies 204 (2006) · Zbl 1092.45003
[10] Kiryakova, V.: Generalized fractional calculus and applications. Pitman research notes in mathematics series 301 (1994)
[11] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations. (1993) · Zbl 0789.26002
[12] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications. (1993)
[13] R.P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for fractional differential equations, Georgian. Math. J. (in press) · Zbl 1179.26011
[14] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. (in press) · Zbl 1198.26004
[15] Belarbi, A.; Benchohra, M.; Hamani, S.; Ntouyas, S. K.: Perturbed functional differential equations with fractional order. Commun. appl. Anal. 11, No. 3--4, 429-440 (2007) · Zbl 1148.34042
[16] Belarbi, A.; Benchohra, M.; Ouahab, A.: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Appl. anal. 85, 1459-1470 (2006) · Zbl 1175.34080
[17] Benchohra, M.; Graef, J. R.; Hamani, S.: Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions. Appl. anal. 87, No. 7, 851-863 (2008) · Zbl 1198.26008
[18] Benchohra, M.; Hamani, S.; Ntouyas, S. K.: Boundary value problems for differential equations with fractional order. Surv. math. Appl. 3, 1-12 (2008) · Zbl 1157.26301
[19] 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, No. 2, 1340-1350 (2008) · Zbl 1209.34096
[20] Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.: Existence results for fractional functional differential inclusions with infinite delay and application to control theory. Fract. calc. Appl. anal. 11, No. 1, 35-56 (2008) · Zbl 1149.26010
[21] Delbosco, D.; Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. math. Anal. appl. 204, 609-625 (1996) · Zbl 0881.34005
[22] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations. J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003
[23] Diethelm, K.; Walz, G.: Numerical solution of fractional order differential equations by extrapolation. Numer. algorithms 16, 231-253 (1997) · Zbl 0926.65070
[24] Kilbas, A. A.; Marzan, S. A.: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differential equations 41, 84-89 (2005) · Zbl 1160.34301
[25] 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 dyn. 29, 281-296 (2002)
[26] Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional diffrential equations. Electron. J. Differential equations 36, 1-12 (2006)
[27] Heymans, N.; Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann--Liouville fractional derivatives. Rheol. acta 45, No. 5, 765-772 (2006)
[28] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. calc. Appl. anal. 5, 367-386 (2002) · Zbl 1042.26003
[29] Benchohra, M.; Henderson, J.; Ntouyas, S. K.: Impulsive differential equations and inclusions. 2 (2006) · Zbl 1130.34003
[30] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differntial equations. (1989)
[31] Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations. (1995) · Zbl 0837.34003
[32] Benchohra, M.; Slimani, B. A.: Impulsive fractional differential equations. Electron. J. Differential equations 10, 1-11 (2009) · Zbl 1178.34004
[33] Heikkila, S.; Lakshmikantham, V.: Monotone iterative technique for nonlinear discontinuous differential equations. (1994)
[34] Ladde, G. S.; Lakshmikanthan, V.; Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations. (1985) · Zbl 0658.35003
[35] Aubin, J. P.; Frankowska, H.: Set-valued analysis. (1990) · Zbl 0713.49021
[36] Deimling, K.: Multivalued differential equations. (1992) · Zbl 0760.34002
[37] Hu, Sh.; Papageorgiou, N.: Handbook of multivalued analysis, theory I. (1997) · Zbl 0887.47001
[38] Kisielewicz, M.: Differential inclusions and optimal control. (1991) · Zbl 0731.49001
[39] Bohnenblust, H. F.; Karlin, S.: On a theorem of ville. Contribution to the theory of games. Annals of mathematics studies 24, 155-160 (1950) · Zbl 0041.25701
[40] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008