Viscosity approximation methods for split common fixed point problems without prior knowledge of the operator norm

Abstract

In this work, we study the split common fixed point problem which was first introduced by Censor and Segal [14]. We introduce an algorithm based on the viscosity approximation method without prior knowledge of the operator norm by selecting the stepsizes in the same adaptive way as L\'{o}pez et al. [22] for solving the problem for two attracting quasi-nonexpansive operators in real Hilbert spaces. A strong convergence result of the proposed algorithm is established under some suitable conditions. We also modify our algorithm to extend to the class of demicontractive operators and the class of hemicontractive operators, and obtain strong convergence results.  Moreover, we apply our main result to other split problems, that is, the split feasibility problem and the split variational inequality problem. Finally, a numerical result is also given to illustrate the convergence of our algorithm.