Solving the General Split Common Fixed-Point Problem of Quasi-Nonexpansive Mappings without Prior Knowledge of Operator Norms

Abstract

Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $A:H_1\rightarrow H_3$, $B:H_2\rightarrow H_3$
be two bounded linear operators. The general split common fixed-point problem under consideration in this paper is to 

$$\text{find}\ \ x \in \cap_{i=1}^p F(U_i),\ \ y \in \cap_{j=1}^r F(T_j)\ \ \text{such that}\ \ Ax = By,\eqno{(1)}$$

 where $p$, $r\geq 1$ are integers, $U_i:H_1\rightarrow H_1$ $(1\leq i\leq p)$

and $T_j:H_2\rightarrow H_2$ $(1\leq j\leq r)$ are
quasi-nonexpansive mappings with nonempty common fixed-point sets $\cap_{i=1}^pF(U_i)=\cap_{i=1}^p\{x\in H_1:U_ix=x\}$ and $\cap_{j=1}^rF(T_j)=\cap_{j=1}^r\{x\in H_2:T_jx=x\}$. Note that, the above problem (1) allows asymmetric and partial relations
between the variables $x$ and $y$. If $H_2=H_3$ and $B=I$, then the general split common fixed-point problem (1) reduces to the general split common fixed-point problem proposed by Censor and Segal $\cite{C}$. In this paper, we introduce simultaneous
parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.