Weak convergence theorems for strongly continuous semigroups of pseudocontractions
Duong Viet Thong
Abstract
Let
K be a nonempty closed convex subset of a uniformly convex Banach space
E, let {
T(
t):
t≥0} be a strongly continuous semigroup of nonexpansive mappings from
K into itself such that
F:=⋂
t≥0
F(
T(
t))≠∅. Assuming that {
α
n
} and {
t
n
} are sequences of real numbers satisfying appropriate conditions, we show that the sequence {
x
n
} defined by
xn=αnxn−1+(1−αn)T(tn)xn
converges weakly to an element of
F. This extends Thong’s result (Thong, Nonlinear Anal. 74, 6116–6120,
2011) from a Hilbert space setting to a Banach space setting. Next, theorems of weak convergence of an implicit iterative algorithm with errors for treating a strongly continuous semigroup of Lipschitz pseudocontractions are established in the framework of a real Banach space.