Approximation methods for nonlinear operator equations

Abstract

Let E be a real normed linear space and A : E → E be a uniformly quasi-accretive map. For arbitrary x 1 E E define the sequence x n E E by x n+1 := x n - α n Ax n , n > 1, where {an} is a positve real sequence satisfying the following conditions: (i) Σα n = ∞; (ii) lim α n = 0. For x* E N(A):= {x E E: Ax = 0}, assume that σ:= inf n ∈ N0 φ(∥x n+1 -x*∥)/∥x n+1 - x*∥ > 0 and that ∥Ax n+1 - Ax n ∥ → 0, where No:= {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.