© Hindawi Publishing Corp. ON THE EQUIVALENCE OF MANN AND ISHIKAWA ITERATION METHODS

We show that certain Mann and Ishikawa iteration schemes are equivalent for various classes of functions. 2000 Mathematics Subject Classification: 47H10. The Mann iterative scheme was invented in 1953, see [7], and was used to obtain convergence to a fixed point for many functions for which the Banach principle fails. For example, the first author in [8] showed that, for any contin-uous selfmap of a closed and bounded interval, the Mann iteration converges to a fixed point of the function. In 1974, Ishikawa [5] devised a new iteration scheme to establish conver-gence for a Lipschitzian pseudocontractive map in a situation where the Mann iteration process failed to converge. Let X be a Banach space. The Mann iteration is defined by x0 ∈X, xn+1