On the Equivalence of Mann and Ishikawa Iteration Methods

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 continuous 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 convergence 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 = ( 1−αn ) xn+αnTxn, n≥ 0, (1)