General local convergence theory for a class of iterative processes and its applications to Newton's method (Q998977)

General local convergence theorems with order of convergence \(r\geq 1\) are provided for iterative processes of type \( x_{n+1}=Tx_{n}\), where \(T:D \subset X \rightarrow X\) is an iteration function in a metric space \(X\). The new local convergence theory is applied to Newton iteration for simple zeros of nonlinear operators in Banach spaces as well as to Schröder iteration for multiple zeros of polynomials and analytic functions.The theory is also applied to establish a general theorem for the uniqueness ball of nonlinear equation in Banach spaces.