Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings

Let E be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, K be a nonempty closed convex and bounded subset of E,T:K-->K be an asymptotically nonexpansive mapping with sequence {kn}(n) subset of [1, infinity). Let u is an element of K be fixed, {tn}(n) subset of (0, 1) be such that lim (n-->infinity) t(n)=1, t(n)k(n)<1, and lim (n ->infinity) k(n)-1/k(n)-t(n) = 0. Define the sequence {zn}(n) iteratively by z(0) is an element of K, z(n+1)=(1-t(n)/k(n))u + t(n)/k(n)T(n)z(n), n=0,1,2,.... It is proved that, for each integer n >= 0, there is a unique x(n) is an element of K such that x(n)=(1-t(n)/k(n))u = t(n)/k(n)T(n)x(n). If, in addition, lim (n ->infinity) parallel to x(n)-Tx(n)parallel to = 0 and lim (n ->infinity) parallel to z(n)-Tz(n)parallel to=0, then {z(n)}(n) converges strongly to a fixed point of T.