Numerical Analysis and Applications

, Volume 9, Issue 4, pp 277–287 | Cite as

On iterative methods for solving equations with covering mappings

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Abstract

In this paper we propose an iterative method for solving the equation Υ(x, x) = y, where the mapping Υ acts in metric spaces and is covering in the first argument and Lipschitzian in the second one. Each subsequent element x i+1 of the sequence of iterations is defined by the previous one as a solution to the equation Υ(x, x i) = y i, where y i can be an arbitrary point sufficiently close to y. Conditions for convergence and error estimates are obtained. The method proposed is an iterative development of the Arutyunov method for finding coincidence points of mappings. In order to determine x i+1 in practical implementation of the method in linear normed spaces, it is proposed to perform one step by using the Newton–Kantorovich method. The thus-obtained method of solving the equation of the form Υ(x, u) = ψ(x) − φ(u) coincides with the iterative method proposed by A.I. Zinchenko,M.A. Krasnosel’skii, and I.A. Kusakin.

Keywords

iterative methods for solving equations covering mappings in metric spaces approximate solution 

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Tambov State Technical UniversityTambovRussia
  2. 2.Derzhavin Tambov State UniversityTambovRussia
  3. 3.Peoples’ Friendship University of RussiaMoscowRussia

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