An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem

Abstract

The multiple-sets split equality problem (MSSEP) requires finding a point $x\in {\cap }_{i=1}^N{C}_i$, $y\in {\cap }_{j=1}^M{Q}_j$ such that $Ax=By$, where $N$ and $M$ are positive integers, $\{C_1,C_2,\dots ,C_N\}$ and $\{Q_1,Q_2,\dots ,Q_M\}$ are closed convex subsets of Hilbert spaces $H_1$, $H_2$, respectively, and $A:H_1\to H_3$, $B:H_2\to H_3$ are two bounded linear operators. When $N=M=1$, the MSSEP is called the split equality problem (SEP). If $B=I$, then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. One of the purposes of this paper is to introduce an iterative algorithm to solve the SEP and MSSEP in the framework of infinite-dimensional Hilbert spaces under some more mild conditions for the iterative coefficient.