A new hybrid iterative algorithm for variational inequalities

摘要

Let H be a real Hilbert space. Let F:H→H be a strongly monotone and Lipschitzian mapping. Let {Tn}n=1∞:H→H be an infinite family of non-expansive mappings with common fixed points set ⋂n=1∞Fix(Tn)≠∅. We devise an iterative algorithmyn=xn-λnF(xn),xn+1=(1-αn)yn+αnWnyn,n⩾0,where {λn} is a sequence in (0,∞), {αn} is a sequence in (0,1) and Wn is the W-mapping. We prove that the sequence {xn} converges in norm to x∗∈⋂n=1∞Fix(Tn) which is the unique solution of the following variational inequality〈Fx∗,x-x∗〉⩾0,∀x∈⋂n=1∞Fix(Tn).