On iterative methods for solving equations with covering mappings
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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.
Keywordsiterative methods for solving equations covering mappings in metric spaces approximate solution
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- 4.Avakov, E.R., Arutyunov, A.V., and Zhukovskiy, E.S., Covering Mappings and Their Application to Differential Equations That Are Not Resolved with Respect to the Derivative, Diff. Ur., 2009, vol. 45, no. 5, pp. 613–634.Google Scholar
- 7.Krasnosel’skii, M.A., Vainikko, G.M., Zabreiko, P.P., Rutitskii, Ya.B., and Stetsenko, V.Ya., Priblizhennoe reshenie operatornykh uravnenii (Approximate Solution of Operator Equations), Moscow: Nauka, 1969.Google Scholar
- 8.Zhukovskaya, T.V. and Molokanova, E.A., Method of Approximate Finding Coincidence Points of Covering and Lipschitzian Mappings, Vest. Tambov Univ. Ser.: Estestv. Tekh. Nauki, 2013, vol. 18, no. 5-2, pp. 2513–2518.Google Scholar