## Convergence to common fixed point of nonexpansive semigroups.(English)Zbl 1204.47076

Summary: Let $$E$$ be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, $$C$$ be a closed convex subset of $$E$$, $$\mathcal S=\{T(s):s\geq 0\}$$ be a nonexpansive semigroup on $$C$$ such that the set of common fixed points of $$\{T(s):s\geq 0\}$$ is nonempty. Let $$f: C\to C$$ be a contraction, $$\{\alpha_n\}$$, $$\{\beta_n\}$$, $$\{t_n\}$$ be real sequences such that $$0<\alpha_n, \beta_n\leq 1$$, $$\lim_{n\to\infty}\alpha_n=0$$, $$\lim_{n\to\infty}\beta_n=0$$ and $$\lim_{n\to\infty}t_n=\infty$$, $$y_0\in C$$. In this paper, we show that the two iterative sequences defined as follows:
\begin{aligned} x_n&=\alpha_nf(x_n)+(1-\alpha_n)\frac 1{t_n}\int_0^{t_n} T(s)x_n\,ds,\\ y_{n-1}&=\beta_nf(y_n)+(1-\beta_n)\frac 1{t_n}\int_0^{t_n} T(s)y_n\,ds, \end{aligned}
converge strongly to a common fixed point of $$\{T(s):s\geq 0\}$$ which solves some variational inequality when $$\{\alpha_n\}$$, $$\{\beta_n\}$$ satisfy some appropriate conditions.

### MSC:

 47H20 Semigroups of nonlinear operators 47J25 Iterative procedures involving nonlinear operators 65J15 Numerical solutions to equations with nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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### References:

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