A general viscosity approximation method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces

Bull. Korean Math. Soc. 2008 Vol. 45, No. 4, 717-728 Printed December 1, 2008

Somyot Plubtieng and Rattanaporn Wangkeeree Naresuan University

Abstract : Let $H$ be a real Hilbert space and $\mathcal{S}=\{T(s): 0\leq s<\infty\}$ be a nonexpansive semigroup on $H$ such that $F(\mathcal{S})\neq \emptyset$. For a contraction $f$ with coefficient $0<\alpha<1$, a strongly positive bounded linear operator $A$ with coefficient $\bar{\gamma}>0$. Let $0<\gamma <\frac{\bar{\gamma}}{\alpha}$. It is proved that the sequences $\{x_t\}$ and $\{x_n\}$ generated by the iterative method $$ x_t=t\gamma f(x_t)+(I-tA)\frac{1}{\lambda_t}\int_0^{\lambda_t}T(s)x_tds,$$ and $$x_{n+1} = \alpha_n\gamma f(x_n) + (I-\alpha_nA)\frac{1}{t_n}\int_0^{t_n}T(s)x_nds,$$
where $\{t\}, \{\alpha_n\}\subset (0,1)$ and $\{\lambda_t\},\{t_n\}$ are positive real divergent sequences, converges strongly to a common fixed point $\tilde{x}\in F(\mathcal{S})$ which solves the variational inequality $\langle (\gamma f-A)\tilde{x},x-\tilde{x}\rangle\leq 0$ for $x\in F(\mathcal{S})$.