Convergence theorems for I-nonexpansive mapping

for all x, y ∈ K . Let F(T) = {x ∈ K : Tx = x} be denoted as the set of fixed points of a mapping T . The first nonlinear ergodic theorem was proved by Baillon [1] for general nonexpansive mappings in Hilbert space : ifK is a closed and convex subset of and T has a fixed point, then for every x ∈ K , {Tnx} is weakly almost convergent, as n→∞, to a fixed point of T . It was also shown by Pazy [7] that if is a real Hilbert space and (1/n) ∑n−1 i=0 Tix converges weakly, as n→∞, to y ∈ K , then y ∈ F(T). The concept of a quasi-nonexpansivemapping was initiated by Tricomi in 1941 for real functions. Diaz andMetcalf [2] and Dotson [3] studied quasi-nonexpansive mappings in Banach spaces. Recently, this concept was given by Kirk [5] in metric spaces which we adapt to a normed space as follows: T is called a quasi-nonexpansive mapping provided