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

Let T:D⊂X→XT:D⊂X→X be an iteration function in a complete metric space XX. In this paper we present some new general complete convergence theorems for the Picard iteration xn+1=Txnxn+1=Txn with order of convergence at least r≥1r≥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 TT and a convergence function   of TT. We study the convergence of the Picard iteration associated to TT with respect to a function of initial conditions E:D→XE:D→X. The initial conditions in our convergence results utilize only information at the starting point x0x0. More precisely, the initial conditions are given in the form E(x0)∈JE(x0)∈J, where JJ is an interval on R+R+ containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ωω-versions of the famous semilocal Newton–Kantorovich theorem as well as a complete version of the famous semilocal αα-theorem of Smale for analytic functions.