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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 ${\germ I}=\{T(s):s\ge 0\}$ 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 $x^*\in F({\germ I})$ which solves the variational inequality $$\langle(A-\gamma f)x^*,x^*-z\rangle\le 0\ z\in F({\germ I}).$$

47J25Iterative procedures (nonlinear operator equations)
47H20Semigroups of nonlinear operators
47J20Inequalities involving nonlinear operators
65J15Equations with nonlinear operators (numerical methods)
Full Text: DOI
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