Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Abstract

Let $H$ be a real Hilbert space. Consider on $H$ a nonexpansive semigroup with a common fixed point, a contraction $f$ with the coefficient , and a strongly positive linear bounded self-adjoint operator $A$ with the coefficient $\overline{\gamma }$> 0. Let /$\alpha$. It is proved that the sequence $\left\{x_n\right\}$ generated by the iterative method $x_0\in H,\hspace{0.17em}{x}_{n+1}={\alpha }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+\left(\right(1-{\beta }_n)I-{\alpha }_{n}A)\left(1/s_n\right){\int }_0^{s_n}T\left(s\right)x_{n}ds,\hspace{0.17em}n\ge 0$ converges strongly to a common fixed point $x^{*}\in F\left(S\right)$, where $F\left(S\right)$ denotes the common fixed point of the nonexpansive semigroup. The point $x^{*}$ solves the variational inequality $\langle (\gamma f-A)x^{*},x-x^{*}\rangle \le 0$ for all $x\in F\left(S\right)$.