Strong Convergence Of Modified Averaging Iterative Algorithm For Asymptotically Nonexpansive Maps. (#39327)

ABSTRACT

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.