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Approximating fixed points of non-self nonexpansive mappings in Banach spaces. (English) Zbl 1089.47058

Let $K$ be a nonempty closed convex subset of a real uniformly convex Banach space $E$, which is also a nonexpansive retract of $E$ $\left(P$ is a nonexpansive retraction of $E$ onto $K\right)$. Let $\left\{{x}_{n}\right\}$ be the sequence defined by ${x}_{1}=x\in K$, ${x}_{n+1}=P\left(\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}TP\left[\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n}\right]\right)$, $n\ge 1$, where $T:E\to K$ is a nonexpansive mapping and $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$ are sequences in $\left[\epsilon ,1-\epsilon \right]$ for some $\epsilon \in \left(0,1\right)$. In the present paper, the author proves the following:

(1) If $F\left(T\right)\ne \varnothing$ and the dual ${E}^{*}$ of $E$ has the Kadec-Klee property, then $\left\{{x}_{n}\right\}$ convergence weakly to some fixed point of $T$.

(2) If $F\left(T\right)\ne \varnothing$ and if there is a nondecreasing function $f:\left[0,+\infty \right)\to \left[0,+\infty \right)$ with $f\left(0\right)=0$ and $f\left(r\right)>0$ for all $r>0$ such that for all $x\in K$, $\parallel x-Tx\parallel \ge f\left(d\left(x,F\left(T\right)\right)\right)$, then $\left\{{x}_{n}\right\}$ converges strongly to some fixed point of $T$.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties