A Real-Time Iterative Projection Scheme for Solving the Common Fixed Point Problem and its Applications

Abstract

In this paper we are concerned with the Common Fixed Point Problem (CFPP) with demi-contractive operators and its special instance, the Convex Feasibility Problem (CFP) in real Hilbert spaces. Motivated by the recent result of Ordoñez et al. [35] and in general, the field of online/real-time algorithms, for example [19, 20, 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 nonlinear feasibility problems with applications to image recovery are presented and demonstrate the validity and potential applicability of our scheme for example to online/real-time scenarios.