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A Cauchy problem for some fractional abstract differential equation with non local conditions. (English) Zbl 1166.34320
The author discusses the existence and uniqueness of a solution to the Cauchy problem for the fractional differential equation with non local conditions $$D^q x(t)=f(t,x(t)),\quad t\in [0,T],\quad x(0)+g(x)=x_0,$$ where $0<q<1$ in a Banach space. Here, the fractional derivative is in the sense of Caputo.

34G20Nonlinear ODE in abstract spaces
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
26A33Fractional derivatives and integrals (real functions)
Full Text: DOI
[1] Aizicovici, S.; Mckibben, M.: Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear anal. TMA 39, 649-668 (2000) · Zbl 0954.34055
[2] Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonloncal Cauchy problem. J. math. Anal. appl. 162, 494-505 (1991) · Zbl 0748.34040
[3] Deng, K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. math. Anal. appl. 179, 630-637 (1993) · Zbl 0798.35076
[4] Ezzinbi, K.; Liu, J.: Nondensely defined evolution equations with nonlocal conditions. Math. comput. Modelling 36, 1027-1038 (2002) · Zbl 1035.34063
[5] Hernández, E.: Existence of solutions to a second order partial differential equation with nonlocal condition. Electron. J. Differential equations 2003, No. 51, 1-10 (2003) · Zbl 1041.35045
[6] V. Lakshmikantham, Theory of fractional differential equations, Nonlinear Anal. TMA, in press (doi:10.1016/j.na.2007.09.025)
[7] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA, in press (doi:10.1016/j.na.2007.08.042)
[8] V. Lakshmikantham, A.S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. (in press) · Zbl 1159.34006
[9] N’guérékata, G. M.: Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions. Differential and difference equations and applications 843--849 (2006) · Zbl 1147.35329
[10] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008