On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces

In this paper, we prove the following strong convergence theorem: Let $C$ be a closed convex subset of a Hilbert space $H$. Let $\{ T(t) : t \geq 0 \}$ be a strongly continuous semigroup of nonexpansive mappings on $C$ such that $\bigcap_{t \geq 0} F\big(T(t)\big) \neq \emptyset$. Let $\{ \alpha_n \}$ and $\{ t_n \}$ be sequences of real numbers satisfying $0 < \alpha_n < 1$, $t_n > 0$ and $\lim_n t_n = \lim_n \alpha_n / t_n = 0$. Fix $u \in C$ and define a sequence $\{ u_n \}$ in $C$ by $u_n = (1 - \alpha_n) T(t_n) u_n + \alpha_n u$ for $n \in \mathbb{N}$. Then $\{ u_n \}$ converges strongly to the element of $\bigcap_{t \geq 0} F\big(T(t)\big)$ nearest to $u$.