Ishikawa and Mann iteration methods for nonlinear strongly accretive mappings

Abstract

Let X be a real Banach space with a uniformly convex dual, X*, and let C be a nonempty closed convex and bounded subset of X. Let T: C → C be a strongly accretive and a continuous mapping. For any f ∈ C, let S: C → C be defined by Sx = f + x – Tx for each x ∈ C. Then, the iteration process xo ∈ C,under suitable conditions on the real sequence converges strongly to a solution of the equation Tx = f in C. Furthermore, if T is strongly accretive and Lipschitz with Lipschitz constant L ≥ 1 then the iteration process x0 ∈ C,under suitable conditions on the real sequences and converges strongly to a solution of the equation Tx = f in C. Explicit error estimates are obtained.