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Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings. (English) Zbl 1194.47089
Summary: Let $K$ be a nonempty closed and convex subset of a real Banach space $E$. Let $T\:K\to E$ be a nonexpansive weakly inward mapping with nonempty fixed point set $\mathrm{Fix}(T)$ and $f\:K\to K$ be a contraction. Then, for $t\in(0,1)$, there exists a sequence $\{y_t\}\subset K$ satisfying $$y_t=(1-t)f(y_t)+tT(y_t).$$ If $E$ is a strictly convex real reflexive Banach space having a uniformly Gâteaux differentiable norm, then $\{y_t\}$ converges strongly to a fixed point $p$ of $T$ such that $p$ is the unique solution in $F(T)$ to a certain variational inequality. Moreover, if $\{T_i\}_{i=1}^r$ is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of $\{T_i\}_{i=1}^r$ and to a solution of a certain variational inequality is constructed. In the above setting, the family $\{T_i\}_{i=1}^r$ is not required to satisfy the condition $$\bigcap_{i=1}^r\mathrm{Fix}(T_i)=\mathrm{Fix}(T_rT_{r-1}\cdots T_1)= \mathrm{Fix}(T_1 T_r\cdots T_2)= \cdots=\mathrm{Fix}(T_{r-1}T_{r-2} \cdots T_1T_r).$$

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties
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##### References:
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