Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings

Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping Jφ for some gauge φ. Let Ti:K→K,i=1,2,…, be a family of quasi-nonexpansive mappings with F≔∩i≥1F(Ti)≠0̸ which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x0∈K, let {xn} be generated by the algorithm xn+1≔αnf(xn)+(1−αn)Tn(xn),n≥0, where f:K→K is a contraction mapping and {αn}⊆(0,1) a sequence satisfying certain conditions. Suppose that {xn}satisfies condition (A). Then it is proved that {xn} converges strongly to a common fixed point x̄=Qf(x̄) of a family Ti,i=1,2,…. Moreover, x̄ is the unique solution in F to a certain variational inequality.