Convergence theorems on viscosity approximation methods for a finite family of nonexpansive non-self-mappings

Let E be a Banach space, C a nonempty closed convex subset of E , f : C → C a contraction, and T i : C → E nonexpansive mappings with nonempty F ≔ ⋂ i = 1 N F i x ( T i ) , where N ≥ 1 is an integer and F i x ( T i ) is the set of fixed points of T i . Assume that C is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. It is proved that an iterative scheme x n + 1 = λ n + 1 f ( x n ) + ( 1 − λ n + 1 ) Q T n + 1 x n ( n ≥ 0 ) converges strongly to a solution in F of a certain variational inequality provided E is strictly convex, reflexive and has a weakly sequentially continuous duality mapping and provided the sequence { λ n } satisfies certain conditions and the sequence { x n } satisfies weak asymptotic regularity. An application of the main result is also given.