CONVERGENCE THEOREMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN BANACH SPACES

Let E be a uniformly convex Banach space, and let K be a nonempty convex closed subset which is also a nonexpansive retract of E. Let T : K ! E be an asymptotically nonexpansive mapping with {kn} (1,1) such that P1 n=1 (kn 1) 0, starting with arbitrary x1 2 K, define the sequence {xn} by setting 8 zn=P( 00T(PT) n1 xn+ 00 nxn+ 00 nwn), yn = P( 0T(PT) n1 zn + 0xn + 0 nvn), xn+1 = P( nT(PT) n 1 yn + nxn + nun), with the restrictions P 1=1 n < 1, P 1=1 0 < 1 and P 1=1 00 < 1, where {wn},{vn} and {un} are bounded sequences in K. (i) If E is real uniformly con- vex Banach space satisfying Opial 0 s condition, then weak convergence of {xn} to some p 2 F(T) is obtained; (ii) If T satisfies condition (A), then {xn} convergence strongly to some p 2 F(T).