## 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.

### MSC:

 34G20 Nonlinear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 26A33 Fractional derivatives and integrals
Full Text:

### References:

 [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, 51, 1-10, (2003) [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, () · Zbl 1169.34331 [10] Podlubny, I., Fractional differential equations, (1999), San Diego Academic Press · Zbl 0918.34010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.