A Monotonicity Condition for Strong Convergence of the Mann Iterative Sequence for Demicontractive Maps in Hilbert Spaces
Akuchu Besheng George, Celestin Akwumbuom Nse · Applied Mathematics · 2014
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 〈Tx, x〉 ≥ ‖x‖2 for all x ∈ D (T). Then the Mann iterative sequence given by xn + 1 = (1 - an) xn + anT xn, where an ∈ (0, 1) n ≥ 0, converges strongly to an element of F (T):= {x ∈ C : Tx = x}. This strong convergence is obtained without the compactness-type assumptions on C, which many previous results (see e.g. [1]) employed.