The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces
Kok-Keong Tan, Hong Kun Xu · Proceedings of the American Mathematical Society · 1992
Let X X be a uniformly convex Banach space with a Frechet differentiable norm, C C a bounded closed convex subset of X X , and T : C → C T:C \to C an asymptotically nonexpansive mapping. It is shown that for each x x in C C , the sequence { T n x } \{ {T^n}x\} is weakly almost-convergent to a fixed point y y of T T , i.e., ( 1 / n ) ∑ i = 0 n − 1 T k + i x → y (1/n)\sum olimits _{i = 0}^{n - 1} {{T^{k + i}}x \to y} weakly as n n tends to infinity uniformly in k = 0 , 1 , 2 , … k = 0,1,2, \ldots