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

Luoyi Shi,Yu Jing Wu,Ru Dong Chen

doi:10.1155/2014/620813

Abstract

The multiple-sets split equality problem (MSSEP) requires finding a pointx∈∩i=1NCi,y∈∩j=1MQjsuch thatAx=By, whereNandMare positive integers,{C1,C2,…,CN}and{Q1,Q2,…,QM}are closed convex subsets of Hilbert spacesH1,H2, respectively, andA:H1→H3,B:H2→H3are two bounded linear operators. WhenN=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.

Highlights

If N = M = 1, the multiple-sets split feasibility problem (MSSFP) reduce to the well-known split feasibility problem (SFP)
The SFP and MSSFP were first introduced by Censor and Elfving [1] and Censor et al [2], respectively, which attract many authors’ attention due to its applications in signal processing [1] and intensity-modulated radiation therapy [2]
Let w be a solution of split equality problem (SEP); according to Lemma 6, we can get that the sequence ‖wn − w‖ is monotonically decreasing and converges to some positive real

Summary

Introduction and Preliminaries

QM} be nonempty closed convex subsets of real Hilbert spaces H1 and H2, respectively, and let A : H1 → H2 be a bounded linear operator. Various algorithms have been invented to solve it; see [1,2,3,4,5,6,7,8], e.t. Recently, Moudafi [9] propose a new split equality problem (SEP): let H1, H2, and H3 be real Hilbert spaces; let C ⊆ H1, Q ⊆ H2 be two nonempty closed convex sets; and let A : H1 → H3, B : H2 → H3 be two bounded linear operators. Let H be a Hilbert space and {wn} a sequence in H such that there exists a nonempty set S ⊆ H satisfying the following.

Iterative Algorithm for SEP

Iterative Algorithm for MSSEP