STRONG CONVERGENCE OF MODIFIED AVERAGING ITERATIVE ALGORITHM FOR ASYMPTOTICALLY NONEXPANSIVE MAPS

Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let
T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set.
Let fng1n
=1 and ftng1 n=1 be real sequences in (0,1). Let fxng be a sequence generated
from an arbitrary x0 2 K by

yn = PK[(1 ? tn)xn]; n  0
xn+1 = (1 ? n)yn + nTnyn; n  0:
where PK : H ! K is the metric projection. Under some appropriate mild conditions
on fng1n
=1 and ftng1 n=1, we prove that fxng converges strongly to xed point of T. No
compactness assumption is imposed on T and or K and no further requirement is imposed
on the xed point set Fix(T) of T