On computational feasibility of successive approximation to a solution of nonlinear operator equations

There are many basic sufficient criteria for theoretical convergence of the successive approximation x_n+1=Tx_n. It should be noted, however, that what is actually obtained is an approximate sequence {y^k} of the exact sequence {x_k} in practical computation because of rounding and discretization errors in the evaluation of T. The question is then, whether or under what condition {y^k} remains convergent if we know {x^k} converges. In this paper, under the assumption that T either is an iterated contraction mapping or satisfies the condition of nonlinear majorants three sufficient criteria for the convergence and feasibility of {y^k} were proposed. These criteria also can be applied to Newton-like iterative method for nonlinear equations.