Suppose that is a real normed linear space, is a nonempty convex subset of , is a Lipschitzian mapping, and is a fixed point of . For given , suppose that the sequence is the Mann iterative sequence defined by , where is a sequence in [0, 1], , . We prove that the sequence strongly converges to if and only if there exists a strictly increasing function with such that .
Chang-He Xiang. Jiang-Hua Zhang. Zhe Chen. "Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings." J. Appl. Math. 2012 (SI15) 1 - 9, 2012. https://doi.org/10.1155/2012/327878