Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings
Chang-He Xiang, Jianghua Zhang, Zhe Chen · Journal of Applied Mathematics · 2012
Suppose that E is a real normed linear space, C is a nonempty convex subset of E , T : C → C is a Lipschitzian mapping, and x * ∈ C is a fixed point of T . For given x 0 ∈ C , suppose that the sequence { x n } ⊂ C is the Mann iterative sequence defined by x n +1 = (1 − α n ) x n + α n T x n , n ≥ 0, where { α n } is a sequence in [0, 1], , . We prove that the sequence { x n } strongly converges to x * if and only if there exists a strictly increasing function Φ : [0, ∞ )→[0, ∞ ) with Φ(0) = 0 such that .