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.
Highlights
Introduction and preliminariesThroughout this paper, we always assume that H is a real Hilbert space with the inner product ·, · and the norm ·
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
1 Introduction and preliminaries Throughout this paper, we always assume that H is a real Hilbert space with the inner product ·, · and the norm ·
Summary
Introduction and preliminariesThroughout this paper, we always assume that H is a real Hilbert space with the inner product ·, · and the norm ·. Where U : H1 → H1 and T : H2 → H2 are two firmly quasi-nonexpansive operators with nonempty fixed-point sets F(U) = {x ∈ H1 : Ux = x} and F(T) = {x ∈ H2 : Tx = x}.
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