## General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space.(English)Zbl 1177.47075

Let $$H$$ be a Hilbert space, $$f$$ a fixed contractive mapping with coefficient $$0<\alpha<1$$, $$A$$ a strongly positive linear bounded operator with coefficient $$\overline\gamma>0$$. Consider two iterative methods that generate the sequences $$\{x_n\}$$, $$\{y_n\}$$ by
$x_n=(1-\alpha_nA)\frac{1} {t_n}\int^{t_n}_0T(s)x_n\, ds+\alpha_n\gamma f(x_n),\tag{I}$
$y_{n+1}=(I-\alpha_n A)\frac{1} {t_n}\int^{t_n}_0T(s)y_n\,ds+\alpha_n\gamma f(y_n),\tag{II}$
where $$\{\alpha_n\}$$ and $$\{t_n\}$$ are two sequences satisfying certain conditions, and $${\mathfrak I}=\{T(s):s\geq 0\}$$ is a one-parameter nonexpansive semigroup on $$H$$. It is proved that the sequences $$\{x_n\}$$, $$\{y_n\}$$ generated by the iterative methods (I) and (II), respectively, converge strongly to a common fixed point $$x^*\in F({\mathfrak I})$$ which solves the variational inequality
$\langle(A-\gamma f)x^*,x^*-z\rangle\leq 0\;z\in F({\mathfrak I}).$

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H20 Semigroups of nonlinear operators 47J20 Variational and other types of inequalities involving nonlinear operators (general) 65J15 Numerical solutions to equations with nonlinear operators
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### References:

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