General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space

Let H be a Hilbert space and f a fixed contractive mapping with coefficient 0 α 1 , A a strongly positive linear bounded operator with coefficient γ > 0 . Consider two iterative methods that generate the sequences { x n } , { y n } by the algorithm, respectively. (I) x n = ( I − α n A ) 1 t n ∫ 0 t n T ( s ) x n d s + α n γ f ( x n ) (II) y n + 1 = ( I − α n A ) 1 t n ∫ 0 t n T ( s ) y n d s + α n γ f ( y n ) where { α n } and { t n } are two sequences satisfying certain conditions, and ℑ = { T ( s ) : s ≥ 0 } is a one-parameter nonexpansive semigroup on H . It is proved that the sequences { x n } , { y n } generated by the iterative method (I) and (II) , respectively, converge strongly to a common fixed point x ∗ ∈ F ( ℑ ) which solves the variational inequality 〈 ( A − γ f ) x ∗ , x ∗ − z 〉 ≤ 0 z ∈ F ( ℑ ) .