Relaxed viscosity approximation methods for fixed point problems and variational inequality problems

Let XX be a real strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm and CC be a nonempty closed convex subset of XX. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on CC such that the common fixed point set F≔⋂n=1∞F(Tn)≠0̸ and f:C→Cf:C→C be a given contractive mapping, and {λn}{λn} be a sequence of nonnegative numbers in [0,1][0,1]. Consider the following relaxed viscosity approximation method {xn+1=(1−αn−βn)xn+αnf(yn)+βnWnyn,yn=(1−γn)xn+γnWnxn,n≥1 where WnWn is the WW-mapping generated by Tn,Tn−1,…,T1Tn,Tn−1,…,T1 and λn,λn−1,…,λ1λn,λn−1,…,λ1 for each n≥1n≥1. It is proven that under very mild conditions on the parameters, the sequence {xn}{xn} of approximate solutions generated by the proposed method converges strongly to some p∈Fp∈F where pp is the unique solution in FF to the following variational inequality: 〈(I−f)p,J(p−x∗)〉≤0,∀x∗∈F.