Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces

Consider on a real Hilbert space H a nonexpansive mapping T with a fixed point, a contraction f with coefficient 0 < α < 1, and two strongly positive linear bounded operators A, B with coefficients over(γ, ̄) ∈ (0, 1) and β > 0, respectively. Let 0 < γ α < β. We introduce a general iterative algorithm defined by x_{n + 1} {colon equals} (I - λ_{n + 1} A) T x_n + λ_{n + 1} [T x_n - μ_{n + 1} (B T x_n - γ f (x_n))], ∀ n ≥ 1, with μ_n → μ (n → ∞), and prove the strong convergence of the iterative algorithm to a fixed point over(x, ̃) ∈ Fix (T) {equals colon} C which is the unique solution of the variational inequality (for short, V I (A - I + μ (B - γ f), C)): 〈 [A - I + μ (B - γ f)] over(x, ̃), x - over(x, ̃) 〉 ≥ 0, ∀ x ∈ C. On the other hand, assume C is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on H. We devise another iterative algorithm which generates a sequence {x_n} from an arbitrary initial point x₀ ∈ H. The sequence {x_n} is proven to converge strongly to an element of C which is the unique solution x^* of the V I (A - I + μ (B - γ f), C). Applications to constrained generalized pseudoinverses are included.