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Convergence on composite iterative schemes for nonexpansive mappings in Banach spaces. (English) Zbl 1203.47053

Let $E$ be a reflexive Banach space with a uniformly Gateaux differentiable norm. Suppose that every weakly compact convex subset of $E$ has the fixed point property for nonexpansive mappings. Let $C$ be a nonempty closed convex subset of $E$, $f:C\to C$ be a contractive mapping (or a weakly contractive mapping), and $T:C\to C$ be a nonexpansive mapping with nonempty fixed point set $F\left(T\right)$. Let the sequence $\left\{{x}_{n}\right\}$ be generated by the following composite iterative scheme:

$\left\{\begin{array}{cc}& {y}_{n}={\lambda }_{n}f\left({x}_{n}\right)+\left(1-{\lambda }_{n}\right)T{x}_{n},\hfill \\ & {x}_{n+1}=\left(1-{\beta }_{n}\right){y}_{n}+{\beta }_{n}T{y}_{n},\hfill \end{array}\right\\phantom{\rule{1.em}{0ex}}n\ge 0·$

It is proved that $\left\{{x}_{n}\right\}$ converges strongly to a point in $F\left(T\right)$, which is a solution of a certain variational inequality, provided that the sequence $\left\{{\lambda }_{n}\right\}\subset \left(0,1\right)$ satisfies ${lim}_{n\to \infty }{\lambda }_{n}=0$ and ${\sum }_{n=1}^{\infty }{\lambda }_{n}=\infty$, $\left\{{\beta }_{n}\right\}\subset \left[0,a\right)$ for some $0, and the sequence $\left\{{x}_{n}\right\}$ is asymptotically regular.

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