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 $$x_n=\alpha_nx_{n-1} +(1-\alpha_n)T(t_n)x_n$$ 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.