Strong Convergence Of A Modified Averaging Iterative Algorithm For Asymptotically Nonexpansive Maps

Abstract

Let C be a nonempty closed convex subset of a real Hilbert space and let T : C → C be an asymptotically nonexpansive mapping with F(T ) = {x ∈ C : T x = x} = ∅. Let {αn}∞n=1, and {tn}∞n=1 be real sequences in (0, 1). Let {xn}∞n=1 be the sequence generated from an arbitrary x1 ∈ C by

νn := PC (1 − tn)xn, n ≥ 1

xn+1 := (1 − αn)νn + αnT nνn, n ≥ 1,
where PC : H → C is the metric projection. Under some appropriate mild conditions on {αn}∞n=1 and {tn}∞n=1, we prove that
{xn}∞n=1 converges strongly to a fixed point of T . No compactness assumption is imposed on T or C and no further requirement is imposed on F(T ).