An approximation method for monotone Lipschitzian operators in Hilbert spaces

Suppose H is a complex Hilbert space and K is a nonempty closed convex subset of H . Suppose T : K → H is a monotomc Lipschitzian mapping with constant L ≧ 1 such that, for x in K and h in H , the equation x + T x Tx = h has a solution q in K . Given x 0 in K , let {C n } ∞ n=0 be a real sequence satisfying: (i) C 0 = 1, (ii) 0 ≦ C n -2 for all n ≧ 1, (iii) Σ n C n (1 − C n ) diverges. Then the sequence {P n } ∞ n=0 in H defined by p n = (1 − C n ) x n + C n Sx n , n ≧ 0, where {x n } ∞ n=0 in K is such that, for each n ≧ 1, ∥ x n – P n−1 ∥ = inf x ∈ k ∥ P n−1 − x ∥ , converges strongly to a solution q of x + Tx = h . Explicit error estimates are given. A similar result is also proved for the case when the operator T is locally Lipschitzian and monotone.