Ergodic Theorem and Strong Convergence of Averaged Approximants for Non-Lipschitzian Mappings in Banach Spaces (original) (raw)

Let C be a bounded closed convex subset of a uniformly convex Banach space X and let T be an asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we first provide a ergodic retraction theorem and a mean ergodic convergence theorem. Using this result, we show that the set F (T ) of fixed points of T is a sunny, nonexpansive retract of C if the norm of X is uniformly Gâteaux differentiable. Moreover, we discuss the strong convergence of the sequence {xn} defined by xn = anx+ (1− an)T (μ)xn for n = 0, 1, 2, . . . , where x ∈ C, μ is a Banach limit on l∞ and an is a real sequence in (0, 1].