Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces

In the present note, using specific uniformly convex Banach spaces techniques of asymptotic center we consider a necessary and sufficient condition for the weak convergence of trajectories of asymptotically nonexpansive mappings. The main result of this paper is contained in the following Theorem: Let E be a uniformly convex Banach space satisfying the Opial 's condition, C a closed convex subset of EtT : C —• C an asymptotically nonexpansive mapping and x € C . Then {Tx} converges weakly to a fixed point of T iff T+x Tx 0 a s n + +00