A Monotonicity Condition for Strong Convergence of the Mann Iterative Sequence for Demicontractive Maps in Hilbert Spaces

Abstract

Let be a real Hilbert space and C be a nonempty closed convex subset of H . Let T : C → C be a demicontractive map satisfying 〈 T x, x〉 ≥ ‖x‖ 2 for all x ∈ D ( T ). Then the Mann iterative sequence given by x n + 1 = (1 - a n ) x n + a n T x n , where a n ∈ (0, 1) n ≥ 0, converges strongly to an element of F ( T ):= {x ∈ C : T x = x}. This strong convergence is obtained without the compactness-type assumptions on C , which many previous results (see e.g. [1]) employed.