An implicit iteration process for nonexpansive semigroups

Abstract

Let C be a closed convex subset of a Banach space E . Let { T ( t ) : t ⩾ 0 } be a strongly continuous semigroup of nonexpansive mappings on C such that ∩ t ⩾ 0 F ( T ( t ) ) ≠ 0̸ . Let { α n } and { t n } be sequences of real numbers satisfying appropriate conditions, then for arbitrary x 0 ∈ C , the Mann type implicit iteration process { x n } given by x n = α n x n − 1 + ( 1 − α n ) T ( t n ) x n , n ⩾ 0 , weakly (strongly) converges to an element of ∩ t ⩾ 0 F ( T ( t ) ) .