Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces (Q5920551)

Let \(K\) be a nonempty closed convex subset of a real Banach space \(E\) and let \(T\) be a self-map of \(K\) such that \(F(T)\), the set of fixed points of \(T\), is nonempty. Then \(T\) is asymptotically quasi-nonexpansive with sequence \(\{v_n\}\subset [0, \infty]\) if \(\lim_{n\rightarrow \infty}= 0\) and \[ \| T^n x - x_*\| \leq\;(1 + v_n) \| x - x_*\| \] for all \(x\in K\), \(x_* \in F(T)\) and \(n \geq 1\). The main purpose of this paper is to investigate conditions under which an Ishikawa type iteration scheme for two asymptotically quasi-nonexpansive self-maps \(S\) and \(T\) of \(K\) converges to a common fixed point of \(S\) and \(T\). The authors consider the case of the Ishikawa type iterations with erors as well. Some results of \textit{M.~K.\ Ghosh} and \textit{L.~Debnath} [J.\ Math.\ Anal.\ Appl.\ 207, No.~1, 96--103 (1997; Zbl 0881.47036)], \textit{Q.--H.\ Liu} [J.\ Math.\ Anal.\ Appl.\ 259, No.~1, 1--7 (2001; Zbl 1033.47047)] and others are discussed as special cases.