Strong Convergence Theorems for Variational Inequalities and Split Equality Problem

Abstract

LetH1, H2,andH3be real Hilbert spaces, letC⊆H1, Q⊆H2be two nonempty closed convex sets, and letA:H1→H3, B:H2→H3be two bounded linear operators. The split equality problem (SEP) is to findx∈C, y∈Qsuch thatAx=By. LetH=H1×H2; considerf:H→Ha contraction with coefficient0<α<1, a strongly positive linear bounded operatorT:H→Hwith coefficientγ̅>0, andM:H→His aβ-inverse strongly monotone mapping. Let0<γ<γ̅/α,S=C×QandG:H→H3be defined by restricting toH1isAand restricting toH2is-B, that is,Ghas the matrix formG=[A,-B]. It is proved that the sequence{wn}={(xn,yn)}⊆Hgenerated by the iterative methodwn+1=PS[αnγf(wn)+(I-αnT)PS(I-γnG*G)PS(wn-λnMwn)]converges strongly tow̃which solves the SEP and the following variational inequality:〈(T-λf)w̃,w-w̃〉≥0and〈Mw̃,w-w̃〉≥0for allw∈S. Moreover, if we takeM=G*G:H→H, γn=0, thenMis aβ-inverse strongly monotone mapping, and the sequence{wn}generated by the iterative methodwn+1=αnγf(wn)+(I-αnT)PS(wn-λnG*Gwn)converges strongly tow̃which solves the SEP and the following variational inequality:〈(T-λf)w̃,w-w̃〉≥0for allw∈S.