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

  • Authors:

  • Petko D. Proinov

  • Affiliations:

  • Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv 4000, Bulgaria

  • Venue:
  • Journal of Complexity
  • Year:
  • 2010

Quantified Score

Hi-index 0.00

Visualization

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=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 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 @w-versions of the famous semilocal Newton-Kantorovich theorem as well as a complete version of the famous semilocal @a-theorem of Smale for analytic functions.