Strong Convergence Theorems for a Finite Family of Nonexpansive Mappings

Let H be a real Hilbert space, C a nonempty closed convex subset of H , and T : C → C a mapping. Recall that T is nonexpansive if ‖Tx−Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. A point x ∈ C is called a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T , that is, F(T) = {x ∈ C : Tx = x}. Recall that a self-mapping f : C → C is a contraction on C, if there exists a constant α∈ (0,1) such that ‖ f (x)− f (y)‖ ≤ α‖x− y‖ for all x, y ∈ C. We use ΠC to denote the collection of all contractions on C, that is, ΠC = { f | f : C→ C a contraction}. An operator A is strongly positive if there exists a constant γ > 0 with the property