A Remark on the Theorem of Ishikawa

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

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.