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A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. (English) Zbl 1198.26004
The paper is a survey on works on the existence and uniqueness of solutions of fractional differential equations with the Caputo fractional derivative. A total of 98 related papers are cited in the references which treat various classes of boundary value problems with local, nonlocal and integral conditions and initial value problems for fractional differential equations and inclusions with impulses. Multivalued cases are also considered and with both convex and nonconvex right hand sides situations. All this is compiled in a nice way in the present survey.

MSC:
26A33Fractional derivatives and integrals (real functions)
34A37Differential equations with impulses
34A60Differential inclusions
34B15Nonlinear boundary value problems for ODE
References:
[1]Adomian, G., Adomian, G.E.: Cellular systems and aging models. Comput. Math. Appl. 11, 283–291 (1985) · Zbl 0565.92017 · doi:10.1016/0898-1221(85)90153-1
[2]Agarwal, R.P., Belmekki, M., Benchohra, M.: Existence results for semilinear functional differential inclusions involving Riemann-Liouville fractional derivative. Dyn. Continuos Discrete Impuls. Syst. (2008, to appear)
[3]Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative (2008, submitted)
[4]Agarwal, R.P., Benchohra, M., Hamani, S.: Boundary value problems for differential inclusions with fractional order. Adv. Stud. Contemp. Math. 16(2), 181–196 (2008)
[5]Agarwal, R.P., Benchohra, M., Hamani, S.: Boundary value problems for fractional differential equations. Georgian Math. J. (2008, to appear)
[6]Agarwal, R.P., Benchohra, M., Slimani, B.A.: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44, 1–21 (2008)
[7]Ahmad, B., Nieto, J.J.: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. (2008). doi: 10.1016/j.na.2007.09.018
[8]Ait Dads, E., Benchohra, M., Hamani, S.: Impulsive fractional differential inclusions involving the Caputo Factional derivative. Fract. Calc. Appl. Anal. (2008, to appear)
[9]Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984)
[10]Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)
[11]Babakhani, A., Daftardar-Gejji, V.: Existence of positive solutions for N-term non-autonomous fractional differential equations. Positivity 9(2), 193–206 (2005) · Zbl 1111.34006 · doi:10.1007/s11117-005-2715-x
[12]Babakhani, A., Daftardar-Gejji, V.: Existence of positive solutions for multi-term non-autonomous fractional differential equations with polynomial coefficients. Electron. J. Differ. Equ. 129, 12 (2006)
[13]Bainov, D.D., Simeonov, P.S.: Systems with Impulsive Effect. Horwood, Chichester (1989)
[14]Belarbi, A., Benchohra, M., Dhage, B.C.: Existence theory for perturbed boundary value problems with integral boundary conditions. Georgian Math. J. 13(2), 215–228 (2006)
[15]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 · doi:10.1080/00036810601066350
[16]Belarbi, A., Benchohra, M., Hamani, S., Ntouyas, S.K.: Perturbed functional differential equations with fractional order. Commun. Appl. Anal. 11(3–4), 429–440 (2007)
[17]Belmekki, M., Benchohra, M.: Existence results for Fractional order semilinear functional differential equations. Proc. A. Razmadze Math. Inst. 146, 9–20 (2008)
[18]Belmekki, M., Benchohra, M., Gorniewicz, L.: Semilinear functional differential equations with fractional order and infinite delay. Fixed Point Theory 9(2), 423–439 (2008)
[19]Benchohra, M., Hamani, S.: Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative. Topol. Methods Nonlinear Anal. (2008, to appear)
[20]Benchohra, M., Hamani, S.: Boundary value problems for differential inclusions with fractional order. Discuss. Math. Differ. Incl. Control Optim. (2008, to appear)
[21]Benchohra, M., Slimani, B.A.: Impulsive fractional differential equations (2008, submitted)
[22]Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi, New York (2006)
[23]Benchohra, M., Hamani, S., Henderson, J.: Functional differential inclusions with integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 15, 1–13 (2007)
[24]Benchohra, M., Hamani, S., Nieto, J.J.: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions. Rocky Mountain J. Math. (2008, to appear)
[25]Benchohra, M., Hamani, S., Nieto, J.J., Slimani, B.A.: Existence results for differential inclusions with fractional order and impulses (2008, submitted)
[26]Benchohra, M., Graef, J.R., Hamani, S.: Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 87(7), 851–863 (2008) · Zbl 1198.26008 · doi:10.1080/00036810802307579
[27]Benchohra, M., Hamani, S., Ntouyas, S.K.: boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008)
[28]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 · doi:10.1016/j.jmaa.2007.06.021
[29]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(1), 35–56 (2008)
[30]Blayneh, K.W.: Analysis of age structured host-parasitoid model. Far East J. Dyn. Syst. 4, 125–145 (2002)
[31]Braverman, E., Zhukovskiy, S.: The problem of a lazy tester, or exponential dichotomy for impulsive differential equations revisited. Nonlinear Anal.: Hybrid Syst. 2, 971–979 (2008) · Zbl 1217.34091 · doi:10.1016/j.nahs.2008.04.002
[32]Bressan, A., Colombo, G.: Extensions and selections of maps with decomposable values. Stud. Math. 90, 70–85 (1988)
[33]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 · doi:10.1016/0022-247X(91)90164-U
[34]Byszewski, L.: Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal Cauchy problem. In: Selected Problems of Mathematics. 50th Anniv. Cracow Univ. Technol. Anniv. Issue, vol. 6, pp. 25–33. Cracow Univ. Technol., Krakow (1995)
[35]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 · doi:10.1080/00036819008839989
[36]Burton, T.A., Kirk, C.: A fixed point theorem of Krasnoselskii-Schaefer type. Math. Nachr. 189, 23–31 (1998) · Zbl 0896.47042 · doi:10.1002/mana.19981890103
[37]Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977)
[38]Chang, Y.-K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. (2008, in press)
[39]Covitz, H., Nadler, S.B. Jr.: Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8, 5–11 (1970) · Zbl 0192.59802 · doi:10.1007/BF02771543
[40]Daftardar-Gejji, V., Jafari, H.: Boundary value problems for fractional diffusion-wave equation. Aust. J. Math. Anal. Appl. 3(1), 8 (2006). Art. 16 (electronic)
[41]Daftardar-Gejji, V., Jafari, H.: Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives. J. Math. Anal. Appl. 328(2), 1026–1033 (2007) · Zbl 1115.34006 · doi:10.1016/j.jmaa.2006.06.007
[42]Dai, B., Su, H., Hu, D.: Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse. Nonlinear Anal. (2008, in press)
[43]Deimling, K.: Multivalued Differential Equations. De Gruyter, Berlin (1992)
[44]Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996) · Zbl 0881.34005 · doi:10.1006/jmaa.1996.0456
[45]Dhage, B.C.: Multivalued mappings and fixed points II. Tamkang J. Math. 37(1), 27–46 (2006)
[46]Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002) · doi:10.1006/jmaa.2000.7194
[47]Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In: Keil, F., Mackens, W., Voss, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217–224. Springer, Heidelberg (1999)
[48]Diethelm, K., Walz, G.: Numerical solution of fractional order differential equations by extrapolation. Numer. Algorithms 16, 231–253 (1997) · Zbl 0926.65070 · doi:10.1023/A:1019147432240
[49]El-Sayed, A.M.A.: Fractional order evolution equations. J. Fract. Calc. 7, 89–100 (1995)
[50]El-Sayed, A.M.A.: Fractional order diffusion-wave equations. Int. J. Theor. Phys. 35, 311–322 (1996) · Zbl 0846.35001 · doi:10.1007/BF02083817
[51]El-Sayed, A.M.A.: Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33, 181–186 (1998) · Zbl 0934.34055 · doi:10.1016/S0362-546X(97)00525-7
[52]Frigon, M., Granas, A.: Théorèmes d’existence pour des inclusions différentielles sans convexité. C. R. Acad. Sci. Paris, Ser. I 310, 819–822 (1990)
[53]Fryszkowski, A.: Fixed Point Theory for Decomposable Sets. Topological Fixed Point Theory and Its Applications, vol. 2. Kluwer Academic, Dordrecht (2004)
[54]Furati, K.M., Tatar, N.-E.: An existence result for a nonlocal fractional differential problem. J. Fract. Calc. 26, 43–51 (2004)
[55]Furati, K.M., Tatar, N.-E.: Behavior of solutions for a weighted Cauchy-type fractional differential problem. J. Fract. Calc. 28, 23–42 (2005)
[56]Gaul, L., Klein, P., Kempfle, S.: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81–88 (1991) · doi:10.1016/0888-3270(91)90016-X
[57]Glockle, W.G., Nonnenmacher, T.F.: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46–53 (1995) · doi:10.1016/S0006-3495(95)80157-8
[58]Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)
[59]Guo, H., Chen, L.: Time-limited pest control of a Lotka-Volterra model with impulsive harvest. Nonlinear Anal.: Real World Appl. (2008). doi: 10.1016/j.nonrwa.2007.11.007
[60]Hernandez, E., Henriquez, H.R., McKibben, M.A.: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Anal. (2008). doi: 10.1016/j.na.2008.03.062
[61]Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45(5), 765–772 (2006) · doi:10.1007/s00397-005-0043-5
[62]Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
[63]Hu, Sh., Papageorgiou, N.: Handbook of Multivalued Analysis, Theory, vol. I. Kluwer, Dordrecht (1997)
[64]Jiang, G., Lu, Q., Qian, L.: Chaos and its control in an impulsive differential system. Chaos, Solitons Fractals 34, 1135–1147 (2007) · Zbl 1142.93424 · doi:10.1016/j.chaos.2006.04.024
[65]Jiang, G., Lu, Q., Qian, L.: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos, Solitons Fractals 31, 448–461 (2007) · Zbl 1203.34071 · doi:10.1016/j.chaos.2005.09.077
[66]Kaufmann, E.R., Mboumi, E.: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 3, 11 (2007)
[67]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 · doi:10.1007/s10625-005-0137-y
[68]Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
[69]Kiryakova, V.: Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics Series, vol. 301. Longman, Harlow (1994). Copublished in the United States with Wiley, New York (1994)
[70]Kisielewicz, M.: Differential Inclusions and Optimal Control. Kluwer, Dordrecht (1991)
[71]Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
[72]Luo, Z., Nieto, J.J.: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. (2008). doi: 10.1016/j.na.2008.03.004
[73]Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien (1997)
[74]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) · doi:10.1063/1.470346
[75]Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)
[76]Mohamad, S., Gopalsamy, K., Akca, H.: Exponential stability of artificial neural networks with distributed delays and large impulses. Nonlinear Anal.: Real World Appl. 9, 872–888 (2008) · Zbl 1154.34042 · doi:10.1016/j.nonrwa.2007.01.011
[77]Momani, S.M., Hadid, S.B.: Some comparison results for integro-fractional differential inequalities. J. Fract. Calc. 24, 37–44 (2003)
[78]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 · doi:10.1155/S0161171204302231
[79]Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)
[80]Ouahab, A.: Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 69(11), 3877–3896 (2007).
[81]Pei, Y., Li, C., Chen, L., Wang, C.: Complex dynamics of one-prey multi-predator system with defensive ability of prey and impulsive biological control on predators. Adv. Complex Syst. 8, 483–495 (2005) · Zbl 1082.92046 · doi:10.1142/S0219525905000579
[82]Podlubny, I.: Fractional Differential Equation. Academic Press, San Diego (1999)
[83]Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calculus Appl. Anal. 5, 367–386 (2002)
[84]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) · Zbl 1041.93022 · doi:10.1023/A:1016556604320
[85]Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, Yverdon (1993)
[86]Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)
[87]Shen, J.H., Li, J.L.: Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays. Nonlinear Anal.: Real World (2007). doi: 110.1016/j.nonrwa.2007.08.026
[88]Smart, D.R.: Fixed Point Theorems, vol. 66. Cambridge University Press, Cambridge (1980)
[89]Wang, W.B., Shen, J.H., Nieto, J.J.: Permanence periodic solution of predator prey system with Holling type functional response and impulses. Discrete Dyn. Nat. Soc. (2007). doi: 10.1155/2007/81756 · Zbl 1146.37370 · doi:10.1155/2007/81756
[90]Wei, C., Chen, L.: A delayed epidemic model with pulse vaccination. Discrete Dyn. Nat. Soc. (2008). Article ID 746951, 13 pages
[91]Xia, Y.: Positive periodic solutions for a neutral impulsive delayed Lotka-Volterra competition system with the effect of toxic substance. Nonlinear Anal.: Real World Appl. 8, 204–221 (2007) · Zbl 1121.34075 · doi:10.1016/j.nonrwa.2005.07.002
[92]Yan, J., Zhao, A., Nieto, J.J.: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems. Math. Comput. Model. 40, 509–518 (2004) · Zbl 1112.34052 · doi:10.1016/j.mcm.2003.12.011
[93]Yu, C., Gao, G.: Existence of fractional differential equations. J. Math. Anal. Appl. 310, 26–29 (2005) · Zbl 1088.34501 · doi:10.1016/j.jmaa.2004.12.015
[94]Zeng, G.Z., Wang, F.Y., Nieto, J.J.: Complexity of delayed predator-prey model with impulsive harvest and Holling type-II functional response. Adv. Complex Syst. 11, 77–97 (2008) · Zbl 1168.34052 · doi:10.1142/S0219525908001519
[95]Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 36, 1–12 (2006) · Zbl 1134.39008 · doi:10.1155/ADE/2006/90479
[96]Zhang, H., Chen, L.S., Nieto, J.J.: A delayed epidemic model with stage structure and pulses for management strategy. Nonlinear Anal.: Real World (2007). doi: 10.1016/j.nonrwa.2007.05.004
[97]Zhang, H., Xu, W., Chen, L.: A impulsive infective transmission SI model for pest control. Math. Methods Appl. Sci. 30, 1169–1184 (2007) · Zbl 1155.34328 · doi:10.1002/mma.834
[98]Zhou, J., Xiang, L., Liu, Z.: Synchronization in complex delayed dynamical networks with impulsive effects. Phys. A: Stat. Mech. Appl. 384, 684–692 (2007) · doi:10.1016/j.physa.2007.05.060