*(English)*Zbl 1123.47044

This article deals with new, more cumbersome and freakish, approximations ${x}_{n}$ to a fixed point for nonexpansive mappings of a nonempty closed convex subset of a Banach space $E$; the authors consider the case when $E$ is reflexive and has a weakly continuous duality mapping and the norm of $E$ is uniformly Gâteaux differentiable. Moreover, they consider a finite family of nonexpansive mappings $\mathcal{T}=\{{T}_{1},\cdots ,{T}_{r}\}$ of $C$ into itself with nonempty set of common fixed points and a class of $W$-mappings generated by $\mathcal{T}$. These $W$-mappings are defined by chains ${U}_{1}={\alpha}_{1}{T}_{1}+(1-{\alpha}_{1})I$, ${U}_{2}={\alpha}_{2}{T}_{2}{U}_{1}+(1-{\alpha}_{2})I,\cdots ,{U}_{r-1}={\alpha}_{r-1}{T}_{r-1}{U}_{r-2}+(1-{\alpha}_{r})I$, $W={U}_{r}={\alpha}_{r}{T}_{r}{U}_{r-1}+(1-{\alpha}_{r})I$, where $\alpha ,\cdots ,{\alpha}_{r}$ are reals from $[0,1]$. The authors’ approximations are

where ${W}_{n}$ is a sequence of $W$-mappings generated by $\mathcal{T}$ and ${\lambda}_{n}$ is a sequence from $(0,1)$ such that

Theorems on the strong convergence of these approximations are proved. Based on these results, the problem of finding a common fixed point of finitely many mappings is also considered.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47H09 | Mappings defined by “shrinking” properties |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |