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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 $E$, $\{T_i\}_{i=1}^N$ a finite family of asymptotically nonexpansive self-mappings on $K$ with common sequence $\{k_n\}_{n=1}^\infty\subset [1,\infty)$, $\{t_n\}$, $\{s_n\}$ be two sequences in $(0,1)$ such that $s_n+t_n=1$ $(n\ge 1)$ and $f$ be a contraction on $K$. Under suitable conditions on the sequences $\{t_n\}$, $\{s_n\}$, we show the existence of a sequence $\{x_n\}$ satisfying the relation $x_n= (1-\frac{1}{k_n})x_n+ \frac{s_n}{k_n} f(x_n)+ \frac{t_n}{k_n} T_{r_n}^nx_n$ where $n=l_nN+r_n$ for some unique integers $l_n\ge 0$ and $1\le rn\le N$. Further we prove that $\{x_n\}$ converges strongly to a common fixed point of $\{T_i\}_{i=1}^N$, which solves some variational inequality, provided $\|x_n-T_ix_n\|\to 0$ as $n\to\infty$ for $i=1,2,\dots,N$. As an application, we prove that the iterative process defined by $z_0\in K$, $z_{n+1}= (1-\frac{1}{k_n})z_n+ \frac{s_n}{k_n} f(z_n)+ \frac{t_n}{k_n} T_{r_n}^nz_n$, converges strongly to the same common fixed point of $\{T_i\}_{i=1}^N$.

49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
49L25Viscosity solutions (infinite-dimensional problems)
47H09Mappings defined by “shrinking” properties
Full Text: DOI
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