General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space
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 \[ \langle(A-\gamma f)x^*,x^*-z\rangle\leq 0\;z\in F({\mathfrak I}). \]
scientific article; zbMATH DE number 5539183