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Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption. (English) Zbl 1165.49007
Summary: Let $E$ be a real Banach space with a uniformly Gâteaux differentiable norm and which possesses uniform normal structure, $K$ a nonempty bounded closed convex subset of $E$, ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ a finite family of asymptotically nonexpansive self-mappings on $K$ with common sequence ${\left\{{k}_{n}\right\}}_{n=1}^{\infty }\subset \left[1,\infty \right)$, $\left\{{t}_{n}\right\}$, $\left\{{s}_{n}\right\}$ be two sequences in $\left(0,1\right)$ such that ${s}_{n}+{t}_{n}=1$ $\left(n\ge 1\right)$ and $f$ be a contraction on $K$. Under suitable conditions on the sequences $\left\{{t}_{n}\right\}$, $\left\{{s}_{n}\right\}$, we show the existence of a sequence $\left\{{x}_{n}\right\}$ satisfying the relation ${x}_{n}=\left(1-\frac{1}{{k}_{n}}\right){x}_{n}+\frac{{s}_{n}}{{k}_{n}}f\left({x}_{n}\right)+\frac{{t}_{n}}{{k}_{n}}{T}_{{r}_{n}}^{n}{x}_{n}$ where $n={l}_{n}N+{r}_{n}$ for some unique integers ${l}_{n}\ge 0$ and $1\le rn\le N$. Further we prove that $\left\{{x}_{n}\right\}$ converges strongly to a common fixed point of ${\left\{{T}_{i}\right\}}_{i=1}^{N}$, which solves some variational inequality, provided $\parallel {x}_{n}-{T}_{i}{x}_{n}\parallel \to 0$ as $n\to \infty$ for $i=1,2,\cdots ,N$. As an application, we prove that the iterative process defined by ${z}_{0}\in K$, ${z}_{n+1}=\left(1-\frac{1}{{k}_{n}}\right){z}_{n}+\frac{{s}_{n}}{{k}_{n}}f\left({z}_{n}\right)+\frac{{t}_{n}}{{k}_{n}}{T}_{{r}_{n}}^{n}{z}_{n}$, converges strongly to the same common fixed point of ${\left\{{T}_{i}\right\}}_{i=1}^{N}$.
##### MSC:
 49J40 Variational methods including variational inequalities 47J20 Inequalities involving nonlinear operators 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 49L25 Viscosity solutions (infinite-dimensional problems) 47H09 Mappings defined by “shrinking” properties
##### References:
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