Improvement of the rate of convergence estimates for some classes of difference schemes for solving an ill-posed Cauchy problem

A number of difference schemes for solving an ill-posed Cauchy problem in a Banach space are studied. The aim of this paper is finding the rate of convergence estimates for the schemes and the corresponding error estimates in dependence of error levels in initial data. The previously known estimates of convergence rate and the error estimates are improved by an optimal choice of the initial elements of the schemes. The classes of the schemes allowing the further strengthening of these estimates are specified. The results of numerical experiments showing the usefulness of the developed approach to the solution of ill-posed Cauchy problems are discussed.