Approximation methods for nonlinear operator equations

Let E be a real normed linear space and A : E → E be a uniformly quasi-accretive map. For arbitrary x₁ ∈ E define the sequence x_n ∈ E by x_n+i := x_n - α_nAx_n, n ≥1, where {α_n} is a positve real sequence satisfying the following conditions: (i) Σα_n = ∞; (ii) lim α_n = 0. For x* ∈ N (A) := {x ∈ E : Ax = 0}, assume that σ := inf_n∈N0 ψ(∥x_n+1-x*∥)/∥x_n+1-x*∥ > 0 and that ∥Ax_n+1 - Ax_n∥ → 0, where N₀ := {n ∈ N (the set of all positive integers): x_n+1 ≠ x*} and ψ : [0,∞) → [0,∞) is a strictly increasing function with ψ(0) = 0. It is proved that a Mann-type iteration process converges strongly to x*. Furthermore if, in addition, A is a uniformly continuous map, it is proved, without the condition on σ, that the Mann-type iteration process converges strongly to x*. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.