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On global solutions to fractional functional differential equations with infinite delay in Fréchet spaces. (English) Zbl 1228.34015
Summary: We investigate global uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. We shall rely on a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to M. Frigon and A. Granas [Ann. Sci. Math. Qué. 22, No. 2, 161–168 (1998; Zbl 1100.47514)]. The results are obtained by using the $$\alpha$$-resolvent family $$(S_{\alpha }(t))_{t\geq 0}$$ on a complex Banach space $$X$$ combined with the above-mentioned fixed point theorem. As an application, a controllability result with one parameter is also provided to illustrate the theory.

##### MSC:
 34A08 Fractional ordinary differential equations and fractional differential inclusions 47H10 Fixed-point theorems 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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##### References:
 [1] Araya, D.; Lizama, C., Almost automorphic mild solutions to fractional differential equations, Nonlinear anal., 69, 3692-3705, (2008) · Zbl 1166.34033 [2] Bonila, B.; Rivero, M.; Rodriquez-Germa, L.; Trujilio, J.J., Fractional differential equations as alternative models to nonlinear differential equations, Appl. math. comput., 187, 79-88, (2007) · Zbl 1120.34323 [3] Jumarie, G., An approach via fractional analysis to non-linearity induced by coarse-graining in space, Nonlinear anal. RWA, 11, 535-546, (2010) · Zbl 1195.37054 [4] Kosmatov, N., Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear anal., 70, 2521-2529, (2009) · Zbl 1169.34302 [5] Luchko, Y.F.; Rivero, M.; Trujillo, J.J.; Velasco, M.P., Fractional models, nonlocality and complex systems, Comput. math. appl., 59, 1048-1056, (2010) · Zbl 1189.37095 [6] Agarwal, R.P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving riemann – liouville fractional derivative, Adv. difference equ., 2009, (2009), (Article ID 981728, 47 pages) · Zbl 1182.34103 [7] He, J.H., Some applications of nonlinear fractional differential equations and their approximations, Bull. sci. technol., 15, 2, 86-90, (1999) [8] Mainardi, F., Fractional calculus, some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004 [9] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley and Sons, Inc New York · Zbl 0789.26002 [10] Kilbas, A.A.; Srivastava, Hari M.; Trujillo, Juan J., () [11] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010 [12] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives. theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003 [13] Agarwal, R.P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl. math., (2010) · Zbl 1198.26004 [14] Agarwal, R.P.; Zhou, Y.; He, Y., Existence of fractional neutral functional differential equations, Comput. math. appl., 59, 1095-1100, (2010) · Zbl 1189.34152 [15] Ahmad, B.; Nieto, J.J., Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. math. appl, 58, 1838-1843, (2009) · Zbl 1205.34003 [16] Ahmad, B.; Nieto, J.J., Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Bound. value probl., 2009, (2009), 11 pages. Article ID 708576 · Zbl 1167.45003 [17] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equations, J. math. anal. appl., 311, 495-505, (2005) · Zbl 1079.34048 [18] 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, 1340-1350, (2008) · Zbl 1209.34096 [19] Chang, Y.K.; Nieto, J.J., Some new existence results for fractional differential inclusions with boundary conditions, Math. comput. modelling, 49, 605-609, (2009) · Zbl 1165.34313 [20] Lakshmikantham, V., Theory of fractional functional differential equation, Nonlinear anal., 69, 3337-3343, (2008) · Zbl 1162.34344 [21] Lakshmikantham, V.; Devi, J.V., Theory of fractional differential equations in a Banach space, Eur. J. pure appl. math., 1, 1, 38-45, (2008) · Zbl 1146.34042 [22] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal., 69, 2677-2682, (2008) · Zbl 1161.34001 [23] Lakshmikantham, V.; Vatsala, A.S., General uniqueness and monotone iteration technique in fractional differential equations, Appl. math. lett., 21, 828-834, (2008) · Zbl 1161.34031 [24] Mophou, G.M.; N’Guérékata, G.M., Mild solutions for semilinear fractional differential equations, Electron. J. differential equations, 2009, 21, 1-9, (2009) · Zbl 1180.34006 [25] Mophou, G.M.; N’Guérékata, G.M., Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Appl. math. comput., 216, 61-69, (2010) · Zbl 1191.34098 [26] N’Guérékata, G.M., A Cauchy problem for some fractional abstract differential equations with non local conditions, Nonlinear anal., 70, 5, 1873-1876, (2009) · Zbl 1166.34320 [27] N’Guérékata, G.M., Remarks on the paper: existence of mild solutions of some neutral fractional functional evolution equations with infinite delay, Appl. math. comput., 216, 61-69, (2010), Int. J. Evol. Equ., 5(3) (2010), 1-3 · Zbl 1191.34098 [28] Zhang, S., Monotone iterative method for initial value problem involving riemann – liouville fractional derivatives, Nonlinear anal., 71, 2087-2093, (2009) · Zbl 1172.26307 [29] Zhong, Y.; Feng, J.; Li, J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal., 71, 3249-3256, (2009) · Zbl 1177.34084 [30] Balachandran, K.; Dauer, J.P., Controllability of nonlinear systems in Banach spaces: a survey, J. optim. theory appl., 115, 7-28, (2002) · Zbl 1023.93010 [31] Balachandran, K.; Kim, J.H., Remarks on the paper “controllability of second order differential inclusion in Banach spaces” [J. math. anal. appl. 285, 537-550 (2003)], J. math. anal. appl., 324, 746-749, (2006) · Zbl 1116.93019 [32] Benchohra, M.; Ouahab, A., Controllability results for functional semilinear differential inclusions in Fréchet spaces, Nonlinear anal., 61, 405-423, (2005) · Zbl 1086.34062 [33] Chang, Y.K.; Nieto, J.J.; Li, W.S., Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, J. optim. theory appl., 142, 267-273, (2009) · Zbl 1178.93029 [34] Balachandran, K.; Park, J.Y., Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear anal.: hybrid syst., 3, 4, 363-367, (2009) · Zbl 1175.93028 [35] 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, 35-56, (2008) · Zbl 1149.26010 [36] Chen, Y.Q.; Ahu, H.S.; Xue, D., Robust controllability of interval fractional order linear time invariant systems, Signal process., 86, 2794-2802, (2006) · Zbl 1172.94386 [37] Shamardan, A.B.; Moubarak, M.R.A., Controllability and observability for fractional control systems, J. fract. calc., 15, 25-34, (1999) · Zbl 0964.93013 [38] Ouahab, A., Local and global existence and uniqueness results for impulsive functional differential equations with multiple delay, J. math. anal. appl., 323, 456-472, (2006) · Zbl 1121.34084 [39] Baghli, S.; Benchohra, M., Perturbed functional and neutral functional evolution equations with infinite delay in Fréchet spaces, Electron. J. diff. equ., 69, 1-19, (2008) · Zbl 1190.34098 [40] Baghli, S.; Benchohra, M., Global uniqueness results for partial functional and neutral functional evolution equations with infinite delay, Differential integral equations, 23, 1-2, 31-50, (2010) · Zbl 1240.34378 [41] Agarwal, R.P.; Baghli, S.; Benchohra, M., Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Fréchet spaces, Appl. math. optim., 60, 253-274, (2009) · Zbl 1179.93041 [42] Hale, J.; Kato, J., Phase space for retarded equations with infinite dealy, Funkcial. ekvac., 21, 11-41, (1978) · Zbl 0383.34055 [43] Hino, Y.; Murukami, S.; Naito, T., () [44] Frigon, M.; Granas, A., Resultats de type leray – schauder pour des contractions sur des espaces de Fréchet, Ann. sci. math. quebec, 22, 161-168, (1998) · Zbl 1100.47514 [45] Quinn, M.D.; Carmichael, N., An approach to nonlinear control problems using the fixed point methods, degree theory and pseudo-inverses, Numer. funct. anal. optim., 7, 197-219, (1984-1985) · Zbl 0563.93013
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