A rate of convergence and error estimates for difference methods used to approximate solutions to ill-posed Cauchy problems in a Banach space

A class of finite difference methods of solving ill-posed Cauchy problems for abstract linear differential equations with sectorial operators in a Banach space is studied. Under various a priori assumptions on a solution, we establish several time-uniform estimates for the accuracy of finite difference approximations. We also give some estimates for errors caused by perturbations of initial conditions.