The comparison of the convergence speed between Picard, Mann, Ishikawa and two-step iterations in Banach spaces (Q445796)

Let \(E\) be an arbitrary real Banach space, \(D\) a closed convex subset of \(E\), \(T\) a Zamfirescu selfmap of \(D\). The author is concerned with the following two-step iteration procedure defined by \textit{S. Thianwan} [J. Comput. Appl. Math. 224, No. 2, 688--695 (2009; Zbl 1161.65043)]. Let \(x_0 \in D\) and define \(y_n = (1 - b_n)x_n + b_nTx_n\), \(x_{n+1} = (1 - a_n)y_n + a_nTy_n\), \(n \geq 0\), where \(\{a_n\}. \{b_n\} \subset [0, 1]\). The author shows that this two-step iteration converges faster than either the Mann or Ishikawa methods for Zamfirescu operators. He also shows that, for this same class of operators, Picard iteration converges faster than the two-step procedure.