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

Abstract:

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 $u$.