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 J_φ for some gauge φ. Let T_i : K → K, i = 1, 2, ..., be a family of quasi-nonexpansive mappings with F {colon equals} ∩_{i ≥ 1} F (T_i) ≠ 0{combining long solidus overlay} which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x₀ ∈ K, let {x_n} be generated by the algorithm x_{n + 1} {colon equals} α_n f (x_n) + (1 - α_n) T_n (x_n), n ≥ 0, where f : K → K is a contraction mapping and {α_n} ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {x_n} satisfies condition (A). Then it is proved that {x_n} converges strongly to a common fixed point over(x, ̄) = Q f (over(x, ̄)) of a family T_i, i = 1, 2, .... Moreover, over(x, ̄) is the unique solution in F to a certain variational inequality.