Strong convergence of implicit viscosity approximation methods for pseudocontractive mappings in Banach spaces

Abstract

Let C be a nonempty closed convex subset of a uniformly smooth Banach space X. Let f : C → C be a fixed contraction mapping, S : C → C be a nonexpansive mapping and T : C → C be a pseudocontractive mapping. Let {α n }, {β n } and {γ n } be three real sequences in (0, 1) such that α n + β n + γ n ⩽ 1. For arbitrary x 0 ∈ C, the sequence {x n } is generated by It is proven that under some conditions, {x n } converges strongly to a fixed point of T, which solves some variational inequality.