Abstract:
In this paper, we present a theorem on a coincidence point of mappings which extends the Arutyunov theorem. The original version of the Arutyunov theorem guaranteed the existence of a coincidence point for two mappings acting in metric spaces, one of which is $\alpha$-covering and the other is $\beta$-Lipschitz, where $\alpha > \beta.$ This theorem was then extended to mappings acting in $(q_1, q_2)$-quasimetric spaces. In this paper, the problem of the existence of a coincidence point is solved for mappings acting from a $(q_1, q_2)$-quasimetric space to a set equipped with a distance satisfying only the identity condition (the distance vanishes if and only if the points coincide). Under conditions similar to those of the Arutyunov theorem, the existence of a coincidence point is proved. In addition, the questions of convergence of sequences of coincidence points of mappings $\psi_n, \varphi_n$ to the coincidence point $\xi$ of mappings $\psi, \varphi$ are investigated under the convergences $\psi_n(\xi)\to \psi(\xi),$$\varphi_n(\xi)\to \varphi(\xi).$
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