Ishikawa and Mann iterative processes with error for nonlinear strongly accretive operator equations (Q1269690)

Let \(\emptyset\neq K\subseteq X\) be a convex subset of a Banach space \(X\) and \(T: X\to X\). An iteration process \[ x_{n+1}=\alpha_n x_n+\beta_nTy_n+\gamma_nu_n,\quad y_n=\widehat\alpha_nx_n+\widehat\beta_nTx_n+\widehat\gamma_nv_n,\qquad n\in {\mathbb Z}_+ \] is called an Ishikawa iteration sequence with errors, iff \(x_0\in K\), \((u_n)\), \((v_n)\in K^{\mathbb N}\) are bounded and \((\alpha_n)\), \((\widehat\alpha_n)\), \((\beta_n)\), \((\widehat\beta_n)\), \((\gamma_n)\), \((\widehat\gamma_n)\in [0,1]^{\mathbb N}\) satisfy \(\alpha_n+\beta_n+\gamma_n=1=\widehat\alpha_n+\widehat\beta_n+\widehat\gamma_n\). The special case \(\widehat\beta_n+\widehat\gamma_n=0\) is referred to as a Mann iteration sequence with errors. The author establishes conditions in case of a real uniformly smooth Banach space for Ishikawa and Mann iteration sequences to converge strongly if either \(T=I-S+f\) (\(S\) strongly accretive, the limit solves \(Tx=f\)) or \(T\) is strongly pseudocontractive (the limit is a fixed point of \(T\)). The above definitions of the iteration processes improve definitions by \textit{L. Liu} [J. Math. Anal. Appl. 194, 114-125 (1995; Zbl 0872.47031)], and the indicated results extend results by Chidume, Deng and Ding, Ishikawa, Mann, and Zhou.