A general iterative method for an infinite family of nonexpansive mappings

Let HH be a real Hilbert space. Consider the iterative sequence xn+1=αnγf(xn)+βnxn+((1−βn)I−αnA)Wnxn,xn+1=αnγf(xn)+βnxn+((1−βn)I−αnA)Wnxn, where γ>0γ>0 is some constant, f:H→Hf:H→H is a given contractive mapping, AA is a strongly positive bounded linear operator on HH and WnWn is the WW-mapping generated by an infinite countable family of nonexpansive mappings T1,T2,…,Tn,…T1,T2,…,Tn,… and λ1,λ2,…,λn,…λ1,λ2,…,λn,… such that the common fixed points set F≔⋂n=1∞Fix(Tn)≠0̸. Under very mild conditions on the parameters, we prove that {xn}{xn} converges strongly to p∈Fp∈F where pp is the unique solution in FF of the following variational inequality: 〈(A−γf)p,p−x∗〉≤0for allx∗∈F, which is the optimality condition for the minimization problem minx∈F12〈Ax,x〉−h(x).