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

Abstract

Let T : D ⊂ 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 = T x 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 E : 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.