A GENERAL VISCOSITY APPROXIMATION METHOD OF FIXED POINT SOLUTIONS OF VARIATIONAL INEQUALITIES FOR NONEXPANSIVE SEMIGROUPS IN HILBERT SPACES
Somyot Plubtieng, Rattanaporn Wangkeeree · Bulletin of the Korean Mathematical Society · 2008
Let H be a real Hilbert space and S = {T(s) : $0\;{\leq}\;s\; 0. Let 0 < $\gamma$ < $\frac{\bar{\gamma}}{\alpha}$ . It is proved that the sequences { $x_t$ } and { $x_n$ } generated by the iterative method $$x_t\;=\;t{\gamma}f(x_t)\;+\;(I\;-\;tA){\frac{1}{{\lambda}_t}}\;{\int_0}^{{\lambda}_t}\;T(s){x_t}ds,$$ and $$x_{n+1}\;=\;{\alpha}_n{\gamma}f(x_n)\;+\;(I\;-\;{\alpha}_nA)\frac{1}{t_n}\;{\int_0}^{t_n}\;T(s){x_n}ds,$$ where {t}, { ${\alpha}_n$ } $\subset$ (0, 1) and { ${\lambda}_t$ }, { $t_n$ } are positive real divergent sequences, converges strongly to a common fixed point $\tilde{x}\;{\in}\;F(S)$ which solves the variational inequality $\langle({\gamma}f\;-\;A)\tilde{x},\;x\;-\;\tilde{x}{\rangle}\;{\leq}\;0$ for $x\;{\in}\;F(S)$ .