STRONG CONVERGENCE THEOREMS OF ISHIKAWA ITERATION PROCESS WITH ERRORS FOR FIXED POINTS OF LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES
Yen-Cherng Lin, Ngai‐Ching Wong, Jen‐Chih Yao · Taiwanese Journal of Mathematics · 2006
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 Lipschitz continuous mapping. Let {un} and {vn} be bounded sequences in K and {αn} and {βn} be real sequences in [0, 1] satisfying some restrictions. Let {xn} be the sequence generated from an arbitrary x1 ∈ K by the Ishikawa iteration process with errors: yn = (1 − βn)xn + βnTxn + vn, xn+1 = (1−αn)xn +αnTyn +un, n ≥ 1. Sufficient and necessary conditions for the strong convergence {xn} to a fixed point of T is established.