A finite difference scheme on a graded mesh for solving Cauchy problems with a fractional Caputo derivative in a Banach space

We study a well-posed Cauchy problem with a fractional Caputo derivative of the order $\alpha\in(0,1)$ in a Banach space. We build and explore a finite difference scheme on a graded mesh for solving such problems. The stability and accuracy estimates for a proposed finite difference scheme are obtained.