Simultaneous iterative algorithms for the split common fixed-point problem of generalized asymptotically quasi-nonexpansive mappings without prior knowledge of operator norms

Abstract

Let , , be real Hilbert spaces, let , be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms (Trans. Math. Program. Appl. 1:1-11, 2013) with weak convergence for the following split common fixed-point problem: 1 where and are two firmly quasi-nonexpansive operators with nonempty fixed-point sets and . Note that by taking and , we recover the split common fixed-point problem originally introduced in Censor and Segal (J. Convex Anal. 16:587-600, 2009). In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of generalized asymptotically quasi-nonexpansive mappings. To estimate the norm of an operator is a very difficult, if it is not an impossible task. The purpose of this paper is to propose a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information as regards the operator norms. MSC:47H09, 47H10, 47J05, 54H25.