Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces

Let C be a nonempty closed convex subset of real Hilbert space H and S = { T ( s ) : 0 ≤ s < ∞ } be a nonexpansive semigroup on C such that F ( S ) ≠ 0̸ . For a contraction f on C , and t ∈ ( 0 , 1 ) , let x t ∈ C be the unique fixed point of the contraction x ⟼ t f ( x ) + ( 1 − t ) 1 λ t ∫ 0 λ t T ( s ) x d s , where { λ t } is a positive real divergent net. Consider also the iteration process { x n } , where x 0 ∈ C is arbitrary and x n + 1 = α n f ( x n ) + β n x n + ( 1 − α n − β n ) 1 s n ∫ 0 s n T ( s ) x n d s for n ≥ 0 , where { α n } , { β n } ⊂ ( 0 , 1 ) with α n + β n < 1 and { s n } are positive real divergent sequences. It is proved that { x t } and, under certain appropriate conditions on { α n } and { β n } , { x n } converges strongly to a common fixed point of S .