On iterative methods for solving equations with covering mappings
T. V. Zhukovskaia1, E. S. Zhukovskiy2,3
1Tambov State Technical University, 392000, Russian Federation, Tambov, Sovetskaya st., 106, Russia
2Tambov State University named after G.R. Derzhavin, 392000, Russian Federation, Tambov, Internatsional'naya st., 33
3Peoples' Friendship University of Russia, 117198, Russian Federation, Moscow, Mikluho-Maklaya st., 6
Keywords: итерационные методы решения уравнений, накрывающие отображения метрических пространств, приближенное решение, iterative methods for solving equations, covering mappings in metric spaces, approximate solution
Abstract
In this paper we propose an iterative method for solving the equation, where a mapping \Upsilon acts in metric spaces, is covering in the first argument and Lipschitzian in the second one. Each subsequent element of a sequence of iterations is defined by the previous one as a solution to the equation, where can be an arbitrary point sufficiently close to. The conditions for convergence and error estimates have been obtained. The method proposed is an iterative development of the Arutyunov method for finding coincidence points of mappings. In order to determine it is proposed to perform one step using the Newton-Kantorovich method or the practical implementation of the method in linear normed spaces. The obtained method of solving the equation of the form coincides with the iterative method proposed by A.I. Zinchenko, M.A. Krasnosel'skii, I.A. Kusakin.
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