STRONG CONVERGENCE OF PATHS FOR NONEXPANSIVE SEMIGROUPS IN BANACH SPACES

Abstract

Let E be a uniformly convex Banach space with a uni-
formly Gateaux differentiable norm, C be a nonempty closed convex
subset of E and f : C → C be a fixed bounded continuous strong
pseudocontraction with the coefficient α ∈ (0, 1). Let {λt }0<t<1
be a net of positive real numbers such that limt→0 λt = ∞ and
S = {T (s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C such
that F (S) ̸= ∅, where F (S) denotes the set of fixed points of the
semigroup. Then sequence {xt } defined by $s_t=tf(x_t)+(1-t)\frac{1}{\lambda_t}\int_0 ^{\lambda_t}T(s)x_tds$ converges strongly as t → 0 to $\bar{x}$ ∈ F (S), which solves the following variational inequality $⟨(f − I)\bar{x}, p − \bar{x}⟩$ ≤0 for all p ∈ F (S).