Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)2227810.22363/2413-3639-2018-64-4-616-636New ResultsНовые результатыResearch ArticleA Real-Time Iterative Projection Scheme for Solving the Common Fixed Point Problem and Its ApplicationsИтерационная проекционная схема в реальном времени для решения задачи об общей неподвижной точке и ее приложенияGibaliAГибалиАavivg@braude.ac.ilTellerDТеллерДktui619@gmail.comOrt Braude College15122018644Contemporary Problems in Mathematics and PhysicsСовременные проблемы математики и физики61663629112019Copyright ©; 2019, Contemporary Mathematics. Fundamental DirectionsCopyright ©; 2019, Современная математика. Фундаментальные направления2019Contemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направленияhttps://journals.rudn.ru/CMFD/article/view/22278In this paper, we are concerned with the Common Fixed Point Problem (CFPP) with demicontractive operators and its special instance, the Convex Feasibility Problem (CFP) in real Hilbert spaces. Motivated by the recent result of Ordon˜ ez et al. [35] and in general, the field of online/real-time algorithms, e.g., [20, 21, 30], in which the entire input is not available from the beginning and given piece-by-piece, we propose an online/real-time iterative scheme for solving CFPPs and CFPs in which the involved operators/sets emerge along time. This scheme is capable of operating on any block, for any finite number of iterations, before moving, in a serial way, to the next block. The scheme is based on the recent novel result of Reich and Zalas [37] known as the Modular String Averaging (MSA) procedure. The convergence of the scheme follows [37] and other classical results in the fields of fixed point theory and variational inequalities, such as [34]. Numerical experiments for linear and non-linear feasibility problems with applications to image recovery are presented and demonstrate the validity and potential applicability of our scheme, e.g., to online/real-time scenarios.В этой работе мы рассматриваем задачу об общей неподвижной точке (CFPP) с демисжимающими операторами и ее частный случай, выпуклую задачу о допустимости (CFP) в вещественных гильбертовых пространствах. Руководствуясь недавними результатами, полученными Ордонесом и др. в работе [35] и в области алгоритмов в реальном времени в общем, например, в [20, 21, 30], где с самого начала нам недоступны целые наборы операторов/множеств, которые затем получаются постепенно, мы предлагаем итерационную схему в реальном времени для решения задач об общей неподвижной точке (CFPP) и выпуклых задач о допустимости (CFP), в которой участвующие операторы/множества появляются со временем. Такая схема способна работать с любыми блоками данных и для любого конечного числа итераций с последовательным переходом к следующему блоку. Схема основана на недавнем результате, описанном в работе Райха и Заласа [37] и известном как процедура модулярного строкового усреднения (MSA). Сходимость схемы следует из [37] и других классических результатов в теории неподвижных точек и области вариационных неравенств, например, [34]. 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