Hybrid viscosity approximation method for zeros of m-accretive operators in banach spaces

Let X be a uniformly smooth Banach space and A be an m-accretive operator on X with A ^-1 (0) ≠ θ. Assume that F: X → X is δstrongly 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 byx _t,n = tf(x _t,n + (1-t)[Jr _nx _t,n-θ F(Jr _nx _t,n] where f: X → X is a contractive map, {r _n}⊂[ε,∞) for some ε > 0 and {θ:t∈⊂(0, 1)}[0, 1) with lim _t→0θ _t/t=0. 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 {y _n= α _nx _n+(1 -α _n)Jr _n,x _n+1 = β _nf (x _n)+(1-β _n) [Jr _n -y _n-λμ _nμ _nF(Jr _n y _n)], ∀n ≥ 0. 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.