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
\[ \langle(A-\gamma f)x^*,x^*-z\rangle\leq 0\;z\in F({\mathfrak I}). \]


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
Full Text: DOI


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