Hybrid Viscosity Approximation Method for Zeros of m-Accretive Operators in Banach Spaces

Abstract

Let X be a uniformly smooth Banach space and A be an m-accretive operator on X with A −1(0) ≠ ∅. Assume that F: X → X is δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1. This article proposes hybrid viscosity approximation methods which combine viscosity approximation methods with hybrid steepest-descent methods. For each t ∈ (0, 1) and each integer n ≥ 0, let {x t, n } be defined by x t, n = tf(x t, n ) + (1 − t)[J r n x t, n − θ t F(J r n x t, n )] where f: X → X is a contractive map, {r n } ⊂ [ϵ, ∞) for some ϵ > 0 and {θ t : t ∈ (0, 1)} ⊂ [0, 1) with . We deduce that as t → 0, {x t, n } converges strongly to a zero p of A, which is a unique solution of some variational inequality. On the other hand, given a point x 0 ∈ X and given sequences {λ n }, {μ n } in [0, 1], {α n }, {β n } in (0, 1], let the sequence {x n } be generated by It is proven that under appropriate conditions {x n } converges strongly to the same zero p of A. The results presented here extend, improve and develop some very recent theorems in the literature to a great extent.