On the weak convergence theorem for nonexpansive semigroups in Banach spaces

Abstract

Assume that K is a closed convex subset of a uniformly convex Banach space E, and assume that {T(s)}s>0 is a nonexpansive semigroup on K. By using the following implicit iteration sequence {xn} defined by xn=(1−αn)xn−1+αn⋅1tn∫0tnT(s)xnds,∀n≥1, the main purpose of this paper is to establish a weak convergence theorem for the nonexpansive semigroup {T(s)}s>0 in uniformly convex Banach spaces without the Opial property. Our results are different from some recently announced results.