On iterative methods for solving equations with covering mappings

T. V. Zhukovskaia ; E. S. Zhukovskiy

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 19 (2016) no. 4, pp. 357-369

Résumé

In this paper we propose an iterative method for solving the equation $\Upsilon(x,x)=y$, where a mapping $\Upsilon$ acts in metric spaces, is covering in the first argument and Lipschitzian in the second one. Each subsequent element $x_{i+1}$ of a sequence of iterations is defined by the previous one as a solution to the equation $\Upsilon(x,x_i)=y_i$, where $y_i$ can be an arbitrary point sufficiently close to $y$. 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 $x_{i+1}$ 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 $\Upsilon(x,u)=\psi(x)-\phi(u)$ coincides with the iterative method proposed by A. I. Zinchenko, M. A. Krasnosel'skii, I. A. Kusakin.