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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 α-resolvent family (S α (t)) t0 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.
34A08Fractional differential equations
47H10Fixed point theorems for nonlinear operators on topological linear spaces
34K99Functional-differential equations
[1]Araya, D.; Lizama, C.: Almost automorphic mild solutions to fractional differential equations, Nonlinear anal. 69, 3692-3705 (2008) · Zbl 1166.34033 · doi:10.1016/j.na.2007.10.004
[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 · doi:10.1016/j.amc.2006.08.105
[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 · doi:10.1016/j.nonrwa.2009.01.003
[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 · doi:10.1016/j.na.2008.03.037
[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 · doi:10.1016/j.camwa.2009.05.018
[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) · Zbl 1182.34103 · doi:10.1155/2009/981728
[7]He, J. H.: Some applications of nonlinear fractional differential equations and their approximations, Bull. sci. Technol. 15, No. 2, 86-90 (1999)
[8]Mainardi, F.: Fractional calculus, some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics, 291-348 (1997)
[9]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[10]Kilbas, A. A.; Srivastava, Hari M.; Trujillo, Juan J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006)
[11]Podlubny, I.: Fractional differential equations, (1999)
[12]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications, (1993) · 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)
[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 · doi:10.1016/j.camwa.2009.05.010
[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 · doi:10.1016/j.camwa.2009.07.091
[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)
[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 · doi:10.1016/j.jmaa.2005.02.052
[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 · doi:10.1016/j.jmaa.2007.06.021
[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 · doi:10.1016/j.mcm.2008.03.014
[20]Lakshmikantham, V.: Theory of fractional functional differential equation, Nonlinear anal. 69, 3337-3343 (2008) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[21]Lakshmikantham, V.; Devi, J. V.: Theory of fractional differential equations in a Banach space, Eur. J. Pure appl. Math. 1, No. 1, 38-45 (2008) · Zbl 1146.34042 · doi:http://www.ejpam.com/ejpam/index.php/ejpam/article/view/84
[22]Lakshmikantham, V.; Vatsala, A. S.: Basic theory of fractional differential equations, Nonlinear anal. 69, 2677-2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[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 · doi:10.1016/j.aml.2007.09.006
[24]Mophou, G. M.; N’guérékata, G. M.: Mild solutions for semilinear fractional differential equations, Electron. J. Differential equations 2009, No. 21, 1-9 (2009) · Zbl 1179.34002 · doi:emis:journals/EJDE/Volumes/2009/21/abstr.html
[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 · doi:10.1016/j.amc.2009.12.062
[26]N’guérékata, G. M.: A Cauchy problem for some fractional abstract differential equations with non local conditions, Nonlinear anal. 70, No. 5, 1873-1876 (2009) · Zbl 1166.34320 · doi:10.1016/j.na.2008.02.087
[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) · Zbl 1191.34098 · doi:10.1016/j.amc.2009.12.062
[28]Zhang, S.: Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives, Nonlinear anal. 71, 2087-2093 (2009) · Zbl 1172.26307 · doi:10.1016/j.na.2009.01.043
[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 · doi:10.1016/j.na.2009.01.202
[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 · doi:10.1023/A:1019668728098
[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 · doi:10.1016/j.jmaa.2005.11.070
[32]Benchohra, M.; Ouahab, A.: Controllability results for functional semilinear differential inclusions in Fréchet spaces, Nonlinear anal. 61, 405-423 (2005) · Zbl 1086.34062 · doi:10.1016/j.na.2004.12.002
[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 · doi:10.1007/s10957-009-9535-2
[34]Balachandran, K.; Park, J. Y.: Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear anal.: hybrid syst. 3, No. 4, 363-367 (2009) · Zbl 1175.93028 · doi:10.1016/j.nahs.2009.01.014
[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 · doi:10.1016/j.sigpro.2006.02.021
[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 · doi:10.1016/j.jmaa.2005.10.015
[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 · doi:emis:journals/EJDE/Volumes/2008/69/abstr.html
[40]Baghli, S.; Benchohra, M.: Global uniqueness results for partial functional and neutral functional evolution equations with infinite delay, Differential integral equations 23, No. 1–2, 31-50 (2010)
[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 · doi:10.1007/s00245-009-9073-1
[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.: Functional differential equations with unbounded delay, Lecture notes in mathematics 1473 (1991) · Zbl 0732.34051
[44]Frigon, M.; Granas, A.: Resultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. sci. Math. Québec 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 · doi:10.1080/01630568508816189