Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces

A General Viscosity Approximation Method of Fixed Point Solutions of Variational Inequalities for Nonexpansive Semigroups in Hilbert Spaces

“…The purpose of this paper is to prove the strong convergence of the continuous scheme {x t } defined by (1.10) and the iterative scheme {x n } defined by (1.12) in a real Hilbert space. The results presented in this paper extend and improve the corresponding ones announced by Shioji and Takahashi [10] and Shimizu and Takahashi [9], and others.…”

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

“…The following theorem is a corollary of Theorem 8 in [9]. Theorem 2 (Shioji and Takahashi [9]). Let C be a closed convex subset of a Hilbert space H. Let {T (t) : t ∈ R + } be a strongly continuous semigroup of nonexpansive mappings on C such that F (T ) = ∅.…”

“…We know that F (T ) is nonempty if C is bounded; see [2]. The following theorem is a corollary of Theorem 8 in [9]. Theorem 2 (Shioji and Takahashi [9]).…”

On the equivalence between some kinds of implicit and explicit iterations with applications

“…By the assumption that T := {T(t) : t ≥ 0} : C → C is a nonexpansive semiproup, hence for each n ≥ 1, S n := ∞ 0 σ n (t)T(t)dt : C → C is a nonexpansive mapping. From [ [19], Theorem 10], we know that (H, C, {S n }, {a n }, P) and (H, C, {S n }, {b n }, P) have the Browder's and Halpern's property, respectively. Hence the conclusions of Theorem 5.2 follow immediately from Theorems 4.3 and 4.5.…”