On the basis of an implicit iterative method for ill-posed operator equations, we introduce a relaxation factor w and a weighted factor μ, and obtain a stationary two-step implicit iterative method. The range of the factors which guarantee the convergence of iteration is explored. We also study the convergence properties and rates for both non-perturbed and perturbed equations. An implementable algorithm is presented by using Morozov discrepancy principle. The theoretical results show that the convergence rates of the new methods always lead to optimal convergent rates which are superior to those of the original one after choosing suitable relaxation and weighted factors. Numerical examples are also given, which coincide well with the theoretical results.
