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Strong convergence theorems for finitely many nonexpansive mappings and applications. (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 $𝒯=\left\{{T}_{1},\cdots ,{T}_{r}\right\}$ of $C$ into itself with nonempty set of common fixed points and a class of $W$-mappings generated by $𝒯$. These $W$-mappings are defined by chains ${U}_{1}={\alpha }_{1}{T}_{1}+\left(1-{\alpha }_{1}\right)I$, ${U}_{2}={\alpha }_{2}{T}_{2}{U}_{1}+\left(1-{\alpha }_{2}\right)I,\cdots ,{U}_{r-1}={\alpha }_{r-1}{T}_{r-1}{U}_{r-2}+\left(1-{\alpha }_{r}\right)I$, $W={U}_{r}={\alpha }_{r}{T}_{r}{U}_{r-1}+\left(1-{\alpha }_{r}\right)I$, where $\alpha ,\cdots ,{\alpha }_{r}$ are reals from $\left[0,1\right]$. The authors’ approximations are

${x}_{n+1}={\lambda }_{n}y+\left(1-{\lambda }_{n}\right){W}_{n}{x}_{n},\phantom{\rule{1.em}{0ex}}n=1,2,\cdots ,\phantom{\rule{4pt}{0ex}}y,{x}_{1}\in C,$

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

$\underset{n\to \infty }{lim}\phantom{\rule{4pt}{0ex}}{\lambda }_{n}=0,\phantom{\rule{1.em}{0ex}}\sum _{n=1}^{\infty }{\lambda }_{n}=\infty ,\phantom{\rule{1.em}{0ex}}\underset{n\to \infty }{lim}\phantom{\rule{4pt}{0ex}}\frac{{\lambda }_{n-1}}{{\lambda }_{n}}=1·$

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