The necessary and sufficient conditions for the qualified convergence of difference methods for approximate solution of the ill-posed Cauchy problem in a Banach space

We study properties of the finite-difference methods for approximate solution of the ill-posed Cauchy problem for a homogeneous equation of the first order with a sectorial operator in a Banach space. We obtain the necessary and sufficient conditions for the qualified (with respect to the step of grid) uniform (on a segment) convergence of approximations to the exact solution of the problem. These conditions represent a priori data about the segment, where a solution exists, or about the sourcewise representation of a certain value of the desired solution.