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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 \({\mathfrak I}=\{T(s):s\geq 0\}\) 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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