Iterative Algorithms for Nonexpansive Mappings

for all x, y ∈ C. We use F T to denote the set of fixed points of T , that is, F T {x ∈ C : Tx x}. Throughout the paper, we assume that F T / ∅. Construction of fixed points of nonlinear mappings is an important and active research area. In particular, iterative methods for finding fixed points of nonexpansive mappings have received a vast investigation; see 1–28 . It is well known that the Picard iteration xn 1 Txn · · · T 1x of the mapping T at a point x ∈ C may, in general, not behave well. This means that it may not converge even in the weak topology. One way to overcome this difficulty is to use Mann’s iteration method that produces a sequence {xn} via the recursive manner: