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The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators. (English) Zbl 1045.47058
It is shown that the convergence of Mann iteration (one-step) is equivalent to the convergence of Ishikawa (two-step) iteration for various classes of non-Lipschitzian operators. Using essentially the technique of this paper, one can prove that the convergence of Mann-Ishikawa iteration is equivalent to the convergence of three-step iterations, which are known as Noor iteraions for non-Lipschitzian operators. Note that Noor iterations [introduced by M. Aslam Noor, J. Math. Anal. Appl. 251, 217–229 (2000; Zbl 0964.49007)] include Mann-Ishikawa iterations as special cases.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H10Fixed point theorems for nonlinear operators on topological linear spaces