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Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications. (English) Zbl 1113.47055

Let \(C\) be a nonempty closed convex subset of a uniformly convex Banach space \(E\). A mapping \(T:C\rightarrow C\) is called uniformly \(L\)-Lipschitzian on \(C\) if, for some \(L>0\), \[ \left\| T^n x-T^n y\right\| \leq L \left\| x-y\right\| \] for all \(x,y \in C\) and for all \(n=1,2,3\dots\). Denote by \(\text{Fix}(T)\) the set of all fixed points of \(T\). If \(\text{Fix}(T)\neq \varnothing\), then \(T\) is called asymptotically quasi-nonexpansive if, for a sequence \(\{k_n\}\subset [0,\infty)\) with \(\lim\limits_{n\rightarrow \infty} k_n=0\), we have \[ \left\| T^n x-p\right\| \leq (1+k_n) \left\| x-p\right\| \] for all \(x \in C\), \(p \in \text{Fix}(T)\) and for all \(n=1,2,3\dots\). It is proved (Theorem 2) that if \(S,T: C\rightarrow C\) are uniformly \(L\)-Lipschitzian and asymptotically quasi-nonexpansive mappings which satisfy a certain condition (A’), then a two-step iteration process of Ishikawa type associated with \(S\) and \(T\) converges to a common fixed point of \(S\) and \(T\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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