Let S be a left amenable semigroup, let 𝒮 = { T ( s ) : s ∈ S } be a representation of S as Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into C with a uniform Lipschitzian condition, let { μ n } be a strongly left regular sequence of means defined on an 𝒮 -stable subspace of l ∞ ( S ) , let f be a contraction on C , and let { α n } , { β n } , and { γ n } be sequences in (0, 1) such that α n + β n + γ n = 1 , for all n . Let x n + 1 = α n f ( x n ) + β n x n + γ n T ( μ n ) x n , for all n ≥ 1 . Then, under suitable hypotheses on the constants, we show that { x n } converges strongly to some z in F ( 𝒮 ) , the set of common fixed points of 𝒮 , which is the unique solution of the variational inequality 〈 ( f − I ) z , J ( y − z ) 〉 ≤ 0 , for all y ∈ F ( 𝒮 ) .