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Acta Mathematica Vietnamica

Weak convergence theorems for strongly continuous semigroups of pseudocontractions

icon-email Duong Viet Thong

Abstract

Let K be a nonempty closed convex subset of a uniformly convex Banach space E, let {T(t):t0} be a strongly continuous semigroup of nonexpansive mappings from K into itself such that F:=T0F(T(t)). Assuming that {αn} and {tn} are sequences of real numbers satisfying appropriate conditions, we show that the sequence xn defined by xn=αnxn1+(1αn)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.