New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems

摘要

Let T:D is contained in X → X be an iteration function in a complete metric space X. In this paper we present some new general complete convergence theorems for the Picard iteration x_(n+1) = Tx_n with order of convergence at least r ≥ 1. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions of T and a convergence function of T. We study the convergence of the Picard iteration associated to T with respect to a function of initial conditions £: D → X. The initial conditions in our convergence results utilize only information at the starting point x_0. More precisely, the initial conditions are given in the form E(x_0) ∈ J, where J is an interval on R_+ containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ω-versions of the famous semilocal Newton-Kantorovich theorem as well as a complete version of the famous semilocal α-theorem of Smale for analytic functions.