On the Strong Convergence of Viscosity Approximation Process for Quasinonexpansive Mappings in Hilbert Spaces

and Applied Analysis 3 But this fails; for example, let us consider the nonexpansive mapping T : R → R defined by Tx −x for all x ∈ R. It is clear that Fix T {0} and 〈x − Tx, x − q〉 2x2 /≥ 4x2 ‖x − Tx‖2. Recall the following identities in a Hilbert space H: for x, y ∈ H, ω ∈ 0, 1 i ‖x y‖2 ‖x‖2 2〈x, y〉 ‖y‖2; ii ‖ 1 −ω x ωy‖2 1 −ω ‖x‖2 ω‖y‖2 − 1 −ω ω‖x − y‖2. The correction of Maingé’s result is as follows. Proposition 1.2. Let C be a subset of a Hilbert space and T : C → C be a mapping with a nonempty fixed-point set Fix T . Suppose that Tω : 1 − ω I ωT where ω ∈ 0, 1 . Then T is quasinonexpansive if and only if 〈 x − Tωx, x − q 〉 ≥ ω 2 ‖x − Tx‖ 1.7 for all x ∈ C and q ∈ Fix T . Proof. Notice that x − Tωx ω x − Tx and ∥ ∥Tωx − q ∥ ∥2 ∥ ∥ Tωx − x x − q ∥ ∥2 ‖Tωx − x‖ 2 〈 Tωx − x, x − q 〉 ∥ ∥x − q∥2 ω2‖x − Tx‖ 2Tωx − x, x − q 〉 ∥ ∥x − q∥2. 1.8