Huang, Jui-Chi Almost stability of iterative procedures for the infinite mappings of uniformly Lipschitzian and asymptotically nonexpansive in the intermediate sense. (English) Zbl 1112.47054 Far East J. Math. Sci. (FJMS) 21, No. 3, 269-284 (2006). Let \(C\) be a (nonempty) closed convex subset of a real Banach space \(E\) and \(T:=(T_i:C\to C\mid i\in \mathbb N)\) be a family of maps such that (a) \(T_i\) is asymptotically nonexpansive in the intermediate sense \(\forall i\in \mathbb N\), (b) \(F(T):=\cap_{i}F(T_i)\neq \emptyset\), and (c) \(T_1\) is strongly successively pseudocontractive with respect to some \(k\in (0,1)\). The convergence (to some \(q\in F(T)\)) and stability of the iterative scheme \(x_{n+1}=(1-\alpha_{n(1)})x_n+\alpha_{n(1)}T_1^nU_{n(2)}x_n\), \(n\geq 1\), where \(U_{n(i)}=(1-\alpha_{n(i)})I+\alpha_{n(i)}T_i^nU_{n(i+1)}\), \(i=1,\dots,n\), are discussed under certain regularity assumptions involving these data. Reviewer: Mihai Turinici (Iaşi) MSC: 47J25 Iterative procedures involving nonlinear operators 47H06 Nonlinear accretive operators, dissipative operators, etc. 47H10 Fixed-point theorems Keywords:Banach space; asymptotically nonexpansive (in the intermediate sense) and strongly (strictly) successively pseudocontractive map; Lipschitz property; iterative scheme; common fixed point; strong convergence; (almost) stable sequence PDFBibTeX XMLCite \textit{J.-C. Huang}, Far East J. Math. Sci. (FJMS) 21, No. 3, 269--284 (2006; Zbl 1112.47054)