mathematicsВестник российских университетов. МатематикаRussian Universities Reports. Mathematics2686-96672782-3342Тамбовский государственный университет имени Г.Р. Державина10.20310/2686-9667-2025-30-152-309-321mathematics-107Research ArticleНАУЧНЫЕ СТАТЬИORIGINAL ARTICLESО точках совпадения в $(q_1, q_2)$-квазиметрическом пространствеOn coincidence points in $(q_1, q_2)$-quasimetric spacehttps://orcid.org/0000-0002-8849-8848БенарабСарраBenarabSarrabenarab.sarraa@gmail.comhttps://orcid.org/0000-0002-3702-0932МерчелаВассимMerchelaWassimmerchela.wassim@gmail.comhttps://orcid.org/0009-0008-0030-6057ХарубиМухаммед ЭльаминKharoubiMohammed E.kharoubi.mohammed.elamin@gmail.comХьяльНауэльKhialNaouelkhialnaouel@gmail.comУниверситет Константины 3 Салах Бубнидер; Лаборатория прикладной математики и моделирования, Университет 8 мая 1945 г.РоссияUniversity Salah Boubnider Constantine 3; Laboratory of Applied Mathematics and Modeling 8 May 1945 UniversityRussian FederationУниверситет Константины 3 Салах Бубнидер; Университет Мустафы Стамбули – Маскара; Лаборатория прикладной математики и моделирования, Университет 8 мая 1945 г.РоссияUniversity Salah Boubnider Constantine 3; Mustapha Stambouli University – Mascara; Laboratory of Applied Mathematics and Modeling 8 May 1945 UniversityRussian FederationЛаборатория прикладной математики и моделирования, Университет 8 мая 1945 г.РоссияLaboratory of Applied Mathematics and Modeling 8 May 1945 UniversityRussian FederationУниверситет Мустафы Стамбули – МаскараРоссияMustapha Stambouli University – MascaraRussian Federation20251212202530152309321Copyright © Бенараб С., Мерчела В., Харуби М.Э., Хьяль Н., 20252025Бенараб С., Мерчела В., Харуби М.Э., Хьяль Н.Benarab S., Merchela W., Kharoubi M.E., Khial N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathematics.elpub.ru/jour/article/view/107

В данной работе мы представляем теорему о точке совпадения отображений, которая обобщает теорему Арутюнова. В первоначальном варианте теоремы Арутюнова гарантируется существование точки совпадения двух отображений, действующих в метрических пространствах, одно из которых является $\alpha$-накрывающим, а другое — $\beta$-липшицевым, причем $\alpha > \beta.$ Затем эта теорема была распространена на отображения, действующие в $(q_1, q_2)$-квазиметрических пространствах. В данной статье задача о существовании точки совпадения решается для отображений, действующих из $(q_1, q_2)$-квазиметрического пространства в множество, снабженное расстоянием, удовлетворяющим лишь условию тождества (расстояние обращается в ноль тогда и только тогда, когда точки совпадают). При условиях, аналогичных предположениям теоремы Арутюнова, доказано существование точки совпадения. Кроме того, исследованы вопросы сходимости последовательностей точек совпадения отображений $\psi_n, \varphi_n$ к точке совпадения $\xi$ отображений $\psi, \varphi$ при сходимости $\psi_n(\xi)\to \psi(\xi),$ $\varphi_n(\xi)\to \varphi(\xi).$

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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The authors declare that there are no conflicts of interest present.