Let E be a real uniformly convex Banach space, and let{Ti : i ∈ I} be N nonexpansive mappings from E into itself with F = {x ∈ E : Tix = x, i ∈ I} ≠ ϕ, where I = {1, 2, …, N}. From an arbitrary initial point x1 ∈ E, hybrid iteration scheme {xn} is defined as follows: xn+1 = αnxn + (1 − αn)(Tnxn − λn+1μA(Tnxn)), n ≥ 1, where A : E → E is an L‐Lipschitzian mapping, Tn = Ti, i = n(mod N), 1 ≤ i ≤ N, μ > 0, {λn}⊂[0, 1), and {αn}⊂[a, b] for some a, b ∈ (0, 1). Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of the mappings {Ti : i ∈ I} are obtained. The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007).