The equivalence between the \(T\)-stabilities of Mann and Ishikawa iterations (Q2488768)

Let \(X\) be a normed space, \(T\) a selfmap of \(X\) and \(x_0\) a point of \(X\). Let (*) \(x_{n+1}=f(T, x_n)\) be an iterative procedure involving the map \(T\). Suppose that the sequence \(\{x_n\}\) converges to a fixed point \(z\) of \(T\). Roughly speaking, the iterative procedure (*) is said to be stable with respect to \(T\) or simply \(T\)-stable if all sequences approximatively close to \(\{x_n\}\) also converge to the fixed point \(z\). For an excellent discussion on the stability of iterative procedures, one may refer to \textit{A. M. Harder} and \textit{T. L. Hicks} [Math. Jap. 33, No. 5, 693--706 (1988; Zbl 0655.47045)] and [\textit{V. Berinde}, ``Iterative approximation of fixed points'' (Efemeride Baia Mare) (2002; Zbl 1036.47037)]. The two well-known iterative procedures for obtaining fixed points of the map \(T\) are the Mann and Ishikawa iterative procedures. The main (surprising) result of this paper states that these two methods of iterations are equivalent from the \(T\)-stability point of view under certain restrictions.