A convergence theorem for Mann fixed point iteration procedure

We establish a general theorem to approximate fixed points of z-operators on a normed space through the Mann iteration process with errors in the sense of Liu [8]. Our result generalizes and improves upon, among others, the corresponding result of Rhoades [12]. Throughout this note, N will denote the set of all positive integers. Let C be a nonempty convex subset of a normed space E and T: C → C be a mapping. The Mann iteration process is defined by the sequence {xn} (see [9]): x1 = x ∈ C, xn+1 = (1 − bn)xn + bnTxn, n ∈ N (1) where {bn} is a sequence in [0, 1]. Liu [8] introduced the concept of Mann iteration process with errors by the sequence {xn} defined as follows: x1 = x ∈ C, xn+1 = (1 − bn)xn + bnTxn + un, n ∈ N. (2) where {bn} is a sequence in [0, 1] and {un} satisfy S∞ n=1 nunn < ∞. This surely contains (1). We recall the following definitions in a metric space (X,d). A mapping T: X → X is called an a-contraction if d(Tx, Ty) ≤ ad(x, y) for all x, y ∈ X, (3) where a ∈ (0, 1). The map T is called Kannan mapping [5] if there exists b ∈ (0, 12) such that d(Tx, Ty) ≤ b[d(x, Tx) + d(y, Ty)] for all x, y ∈ X. (4