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Existence and uniqueness for fractional neutral differential equations with infinite delay. (English) Zbl 1177.34084

Summary: We consider the Cauchy initial value problem of fractional neutral functional differential equations with infinite delay of the form
\[ D^qg(t,x_t)=f(t,x_t),\quad t\in [t_0,\infty),\tag{1} \]
\[ x_{t_0}=\varphi,\;(t_0,\varphi)\in [0,\infty)\times \Omega,\tag{2} \]
where \(D^q\) is Caputo’s fractional derivative of order \(0 < q < 1\), \(\Omega\) is an open subset of \(B\) and \(g,f : [t_0,\infty)\times \Omega\to \mathbb{R}^n\) are given functionals satisfying some assumptions. Various criteria on existence and uniqueness are obtained.

MSC:

34K05 General theory of functional-differential equations
26A33 Fractional derivatives and integrals
34K40 Neutral functional-differential equations
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[1] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., ()
[2] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[3] Podlubny, I., Fractional differential equations, (1993), Academic Press New York · Zbl 0918.34010
[4] R.P. Agarwal, M. Belmekki, M. Benchohra, Existence results for semilinear functional differential inclusions involving Riemann-Liouville derivative, Dyn. Contin. Discrete Impuls. Syst. (in press) · Zbl 1198.34178
[5] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations, Acta Appl. Math. (2009), doi:10.1007/s10440-008-9356-6 · Zbl 1198.26004
[6] B. Ahmad, J.J. Nieto, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Boundary Value Problems (in press) · Zbl 1167.45003
[7] Araya, D.; Lizama, C., Almost automorphic mild solutions to fractional differential equations, Nonlinear anal., 69, 3692-3705, (2008) · Zbl 1166.34033
[8] M. Belmekki, J.J. Nieto, R. Rodriguez-Lopez, Existence of periodic solution for a nonlinear fractional differential equation, Preprint · Zbl 1324.34063
[9] Bonilla, B.; Rivero, M.; Rodriguez-Germa, L.; Trujillo, J.J., Fractional differential equations as alternative models to nonlinear differential equations, Appl. math. comput., 187, 79-88, (2007) · Zbl 1120.34323
[10] Y.K. Chang, J.J. Nieto, Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Numer. Funct. Anal. Optim. (in press) · Zbl 1176.34096
[11] 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
[12] Daftardar-Gejji, V.; Jafari, H., Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. math. anal. appl., 328, 1026-1033, (2007) · Zbl 1115.34006
[13] Daftardar-Gejji, V.; Bhalekar, Sachin, Boundary value problems for multi-term fractional differential equations, J. math. anal. appl., 345, 754-765, (2008) · Zbl 1151.26004
[14] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005
[15] Diethelm, K., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[16] El-Borai, Mahmoud M., Semigroups and some nonlinear fractional differential equations, Appl. math. comput., 149, 823-831, (2004) · Zbl 1046.34079
[17] El-Sayed, A.M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear anal., 33, 181-186, (1998) · Zbl 0934.34055
[18] Gafiychuk, V.; Datsko, B.; Meleshko, V., Mathematical modeling of time fractional reaction – diffusion systems, J. comput. appl. math., 220, 215-225, (2008) · Zbl 1152.45008
[19] Ibrahim, Rabha W.; Momani, Shaher, On the existence and uniqueness of solutions of a class of fractional differential equations, J. math. anal. appl., 334, 1, 1-10, (2007) · Zbl 1123.34302
[20] Jaradat, Omar K.; Al-Omari, Ahmad; Momani, Shaher, Existence of the mild solution for fractional semilinear initial value problems, Nonlinear anal., 69, 9, 3153-3159, (2008) · Zbl 1160.34300
[21] Nickolai Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Anal. (2009), doi:10.1016/j.na.2008.03.037 · Zbl 1169.34302
[22] Lakshmikantham, V.; Vatsala, A.S., General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. lett., 21, 828-834, (2008) · Zbl 1161.34031
[23] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal., 69, 2677-2682, (2008) · Zbl 1161.34001
[24] M. Muslim, Existence and approximation of solutions to fractional differential equations, Math. Comput. Modelling (2009), doi:10.1016/j.mcm.2008.07.013 · Zbl 1165.34304
[25] Salem, H.A.H., On the existence of continuous solutions for a singular system of non-linear fractional differential equations, Appl. math. comput., 198, 445-452, (2008) · Zbl 1153.26004
[26] J. Vasundhara Devi, V. Lakshmikantham, Nonsmooth analysis and fractional differential equations, Nonlinear Anal. (2009), doi:10.1016/j.na.2008.09.003 · Zbl 1237.49022
[27] Lakshmikantham, V., Theory of fractional functional differential equations, Nonlinear anal., 69, 3337-3343, (2008) · Zbl 1162.34344
[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, 1340-1350, (2008) · Zbl 1209.34096
[29] Zhou, Yong, Existence and uniqueness of fractional functional differential equations with unbounded delay, Int. J. dyn. syst. differ. equ., 1, 4, 239-244, (2008) · Zbl 1175.34081
[30] Zhou, Yong; Jiao, Feng; Li, Jing, Existence and uniqueness for \(p\)-type fractional neutral differential equations, Nonlinear anal., 71, 7-8, 2724-2733, (2009) · Zbl 1175.34082
[31] Lakshmikantham, V.; Wen, L.; Zhang, B., Theory of differential equations with unbounded delay, (1994), Kluwer Academic Publishers Dordrecht · Zbl 0823.34069
[32] Hino, Y.; Murakami, S.; Naito, T., ()
[33] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048
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