THREE-STEP ITERATIVE CONVERGENCE THEOREMS WITH ERRORS IN BANACH SPACES

Abstract

Let q > 1 and E be a real q-uniformly smooth Banach space, K be a nonempty closed convex subset of E and T : K → K be a single-valued mapping. Let {un}∞ n=1, {vn}∞ n=1, {wn}∞ n=1 be three sequences in K and {αn}∞ n=1, {βn}∞ n=1 and {γn}∞ n=1 be real sequences in [0, 1] satisfying some restrictions. Let {xn} be the sequence generated from an arbitrary x1 ∈ K by the three-step iteration process with errors: xn+1 = (1−αn)xn+αnTyn+un, yn = (1 − βn)xn + βnTzn + vn, zn = (1 − γn)xn + γnTxn + wn, n ≥ 1. Sufficient and necessary conditions for the strong convergence {xn} to a fixed point of T is established. We also derive the corresponding new results on the strong convergence of the three-step iterative process.