A Remark on the Theorem of Ishikawa

Abstract

Given a Lipschitz pseudocontractive mapping T from a closed convex and bounded subset K of a real Hilbert space H onto itself, and an arbitrary x1 ∈ K, a Krasnolselskii-type sequence defined by xn+1 = (1− λ)xn + λTyn, yn = (1− λ)xn + λTxn is proved to be an approximate fixed point sequence of T , for a suitable λ ∈ (0, 1). Under some suitable compactness assumptions on K or on T , the sequence converges strongly to a fixed point of T . The algorithm is simple and natural, and the theorems presented here improve the theorem of Ishikawa [1] and other similar results in the literature.