Convergence to common fixed point of nonexpansive semigroups

Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, C be closed convex subset of E, S = { T ( s ) : s ⩾ 0 } be a nonexpansive semigroup on C such that the set of common fixed points of { T ( s ) : s ⩾ 0 } is nonempty. Let f : C → C be a contraction, { α n } , { β n } , { t n } be real sequences such that 0 < α n , β n ⩽ 1 , lim n → ∞ α n = 0 , lim n → ∞ β n = 0 and lim n → ∞ t n = ∞ , y 0 ∈ C . In this paper, we show that the two iterative sequence as follows: x n = α n f ( x n ) + ( 1 - α n ) 1 t n ∫ 0 t n T ( s ) x n d s , y n + 1 = β n f ( y n ) + ( 1 - β n ) 1 t n ∫ 0 t n T ( s ) y n d s converge strongly to a common fixed point of { T ( s ) : s ⩾ 0 } which solves some variational inequality when { α n } , { β n } satisfy some appropriate conditions.