Iterative Scheme for Non-Expansive Semigroups in Hilbert Space
Xiufang Liu · 2009
The main result of this paper is that I consider two iterative methods that generate the sequence $\{x_n\},\{y_n\}$ by the algorithm,respectively.$$x_n=(I-\alpha_n)T(t_n)x_n+\alpha_n\gamma f(x_n),\eqno{(1.3)}$$$$y_{n+1}=(I-\alpha_n)T(t_n)y_n+\alpha_n\gamma f(y_n),\eqno{(1.4)}$$where $\{x_n\}$ and $\{t_n\}$ are two sequences satisfying certain conditions and $\Omega =\{S(t):t\geq 0\}$ is a non-expansive semigroup on $H$.It is proved that the sequence $\{x_n\},\{y_n\}$generated by the iterative method $(1.3)$ and $(1.4)$,respectively,converges strongly to a common fixed point $x^{\ast}\in F$ which solves the variational inequality $\langle (A-\gammaf)x^{\ast},x^{\ast}-z\rangle \leq 0,Z\in F(\Omega)$.