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

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