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 〈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.
Share and Cite:
George, A. and Nse, C. (2014) A Monotonicity Condition for Strong Convergence of the Mann Iterative Sequence for Demicontractive Maps in Hilbert Spaces.
Applied Mathematics,
5, 2195-2198. doi:
10.4236/am.2014.515212.
Conflicts of Interest
The authors declare no conflicts of interest.
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