Nonlinear Studies

We are concerned in this paper with the existence results of the Cauchy problem for the fractional differential equation with non local conditions in $\R^n$: $$D^{q}x(t) = Ax(t)+f(t,x(t)),\quad t\in [0,T],\quad x(0)+g(x) = x_0$$ where $0<q<1$.